Abstract group 699840.c: $C_3^6:(C_8:S_5)$

• Label: 699840.c
• Order: \( 2^{6} \cdot 3^{7} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{7} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $54$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$C_3^6:(C_8:S_5)$
Order:	\(699840\)\(\medspace = 2^{6} \cdot 3^{7} \cdot 5 \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_3^6:(C_8:S_5)$, of order \(699840\)\(\medspace = 2^{6} \cdot 3^{7} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$ x 6, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	18	20	24	40
Elements	1	3159	2348	91368	1944	82620	46656	12960	17496	223560	15552	38880	34992	58320	69984	699840
Conjugacy classes	1	4	6	5	1	13	4	1	1	7	1	1	2	2	4	53
Divisions	1	4	6	5	1	13	2	1	1	5	1	1	1	1	1	44
Autjugacy classes	1	4	6	5	1	13	4	1	1	7	1	1	2	2	4	53

Dimension	1	2	4	5	6	8	10	12	16	20	24	48	80	96	120	160	192	240	480
Irr. complex chars.	4	3	4	4	8	3	3	0	0	0	0	4	4	0	4	2	1	9	0	53
Irr. rational chars.	4	1	5	4	2	1	1	1	1	1	1	2	4	1	4	2	1	7	1	44

Minimal presentations

Permutation degree:	$54$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$165984$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	48	48	48
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $54$ $\langle(1,4,30,27,11,3,13,10)(2,19,29,54,14,53,8,52)(5,50)(7,44)(9,43,24,46,36,41,25,42) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^6$ $\,\rtimes\,$ $(C_8:S_5)$	$(C_3^6:A_5)$ $\,\rtimes\,$ $\SD_{16}$	$(C_3^6:A_5:Q_8)$ $\,\rtimes\,$ $C_2$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^6.C_4.S_5)$ . $C_2$	$(C_3^5:S_3)$ . $(C_4:S_5)$	$(C_3^6.C_4)$ . $(C_2\times S_5)$	$(C_3^6:(C_2\times A_5))$ . $D_4$	all 5
Aut. group:	$\Aut(C_3^6:A_5)$	$\Aut(C_3^6:(C_2\times A_5))$	$\Aut(C_2\times C_3^6:A_5)$	$\Aut(C_3^6:(C_4\times A_5))$	all 5

Elements of the group are displayed as permutations of degree 54.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 7223820 subgroups in 2418 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^6:(C_8:S_5)$
Commutator:	$G' \simeq$ $C_3^6:(C_4\times A_5)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^6:(C_8:S_5)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $C_8:S_5$
Radical:	$R \simeq$ $C_3^6.C_8$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_3^6$	$G/\operatorname{soc} \simeq$ $C_8:S_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 699840: $C_3^6:(C_8:S_5)$

Order 349920: $C_3^6.C_8.A_5$, $C_3^6.C_4.S_5$, $C_3^6:A_5:Q_8$

Order 174960: $C_3^6:(C_2\times S_5)$, $C_3^6:(C_4\times A_5)$, $C_3^6:A_5:C_4$

Order 139968: $C_3^6:(C_8:S_4)$

Order 116640: $C_3^6.C_{40}:C_4$

Order 87480: $C_3^6:S_5$, $C_3^6:(C_2\times A_5)$

Order 69984: $C_3^5:D_6:D_{12}$, $C_3^6:(S_3\times \SD_{16})$, $(C_3:S_3)^3.D_6$, $C_3^6:(C_8\times A_4)$

Order 58320: $C_3^6.C_{20}:C_4$, $C_3^4:(D_5\times F_9)$, $C_3^6:(C_{20}:C_4)$

Order 46656: $C_3^6:C_8:D_4$

Order 43740: $C_3^6:A_5$

Order 34992: $C_3^5:S_3.S_4$, $C_3^6.(C_2\times S_4)$, $C_3^6.(S_3\times D_4)$, $C_3^6.(C_4\times A_4)$, $C_3^6.(S_3\times Q_8)$, $C_3^2\wr C_3:\SD_{16}$, $C_3^6:(S_3\times C_8)$, $C_3^2\wr C_3:\SD_{16}$, $C_3^2\wr C_3:\SD_{16}$, $C_3^6:(C_3\times \SD_{16})$

Order 29160: $C_3^6:(C_2\times F_5)$, $C_3^5:(S_3\times F_5)$, $C_3^6:(C_4\times D_5)$, $C_3^5:C_{15}:C_8$, $C_3^6.C_{40}$

Order 23328: $C_3^6.(C_2\times \SD_{16})$, $C_3^6:(C_2^2\times C_8)$, $C_3^6.C_8:C_4$, $C_3^5:C_6.Q_{16}$, $C_3^6.C_4:D_4$, $C_3^5:C_6.D_8$, $C_3^4.C_6^2:Q_8$

Order 17496: $C_3^2\wr C_3:Q_8$ x 2, $C_3^6:(C_4\times S_3)$ x 2, $C_3^2\wr C_3:Q_8$, $C_3^2\wr C_3:C_8$, $C_3^6.C_{24}$, $C_3^6.S_4$, $C_3^2\wr C_3:D_4$, $C_3^6.D_{12}$, $C_3^6.(C_2\times A_4)$, $C_3^4.S_3^3$, $C_3^2\wr C_3:D_4$

Order 14580: $C_3^6:D_{10}$, $C_3^5:C_{15}:C_4$, $C_3^6:F_5$, $C_3^6:F_5$, $C_3^6.C_{20}$

Order 12960: $C_3^4:C_{40}:C_4$

Order 11664: $C_3^6.C_8.C_2$, $C_3^6.\SD_{16}$, $C_3^5:C_6.D_4$, $C_3^6:(C_2\times D_4)$, $C_3^5:C_6.D_4$, $C_3^6:(C_2^2\times C_4)$, $C_3^6.(C_2\times D_4)$, $C_3^6.C_4:C_4$, $C_3^6.\SD_{16}$, $C_3^5:C_{12}:C_4$, $C_3^6.(C_2\times Q_8)$, $C_3^4.(C_2\times F_9)$, $C_3^6.C_4:C_4$

Order 9720: $C_3^4:(S_3\times F_5)$

Order 8748: $C_3^2\wr C_3:C_4$ x 2, $C_3^6.C_{12}$ x 2, $C_3^6.D_6$, $C_3^6.A_4$, $C_3^6:D_6$, $C_3^6.D_6$, $C_3^2\wr C_3:C_2^2$, $C_3\wr D_6$

Order 7776: $C_3^4:(S_3\times \SD_{16})$

Order 7290: $C_3^6:C_{10}$, $C_3^6:D_5$, $C_3^5:D_{15}$

Order 6480: $C_3^4:C_{20}:C_4$ x 2, $C_3^4:(C_8\times D_5)$

Order 5832: $C_3^6:(C_2\times C_4)$ x 3, $C_3^6:Q_8$ x 2, $C_3^6:D_4$ x 2, $C_3^6.C_8$, $(C_3:S_3)^3$, $C_3^6:C_8$, $C_3^6:(C_2\times C_4)$, $C_3^6:(C_2\times C_4)$, $C_3^6:C_2^3$, $C_3^6:D_4$, $C_3^5:D_{12}$, $C_3^3.S_3^3$

Order 5184: $C_3^2:F_9:D_4$

Order 4860: $C_3\wr F_5$, $C_3^5:D_{10}$, $C_3^5:F_5$

Order 4374: $C_3^6:S_3$, $C_3^6.C_6$, $C_3^2\wr S_3$, $C_3\wr S_3$, $C_3\wr C_6$

Order 3888: $C_3^2:F_9:S_3$, $C_3^3:F_9:C_2$, $C_3^4:(C_3\times \SD_{16})$, $C_3^4:(S_3\times Q_8)$, $C_3^3:(C_2\times F_9)$, $C_3^3:F_9:C_2$, $S_3\times C_3^4:D_4$, $C_3^4:(S_3\times D_4)$

Order 3645: $C_3^5:C_{15}$

Order 3240: $C_3^4:(C_2\times F_5)$ x 2, $C_3^4:C_5:C_8$, $C_3^4:(C_4\times D_5)$, $C_3^4:C_{40}$

Order 2916: $C_3^6.C_4$ x 3, $C_3^5.D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^4:C_6^2$, $C_3^5:C_{12}$, $C_3^4:C_6^2$, $C_3^5:C_{12}$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:(C_2\times C_6)$, $C_3^5.D_6$, $C_3^5:D_6$

Order 2592: $C_3^2\wr C_2.\SD_{16}$ x 2, $C_3^4:(C_2\times \SD_{16})$ x 2, $(C_3^3\times C_6).Q_{16}$, $(C_3^3\times C_6).\SD_{16}$, $C_2\times C_3^4:\SD_{16}$, $C_3^2:(C_2^2\times F_9)$, $(C_3^3\times C_6).D_8$, $C_3^4:C_4:D_4$, $C_3^2:S_3^2:Q_8$

Order 2430: $C_3\wr D_5$, $C_3^4:D_{15}$, $C_3^5:C_{10}$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^5:D_4$ x 2, $C_3\wr C_4:S_3$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^4:C_4\times S_3$, $C_3^2.S_3^3$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^4:D_{12}$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$, $C_3^4:C_{24}$

Order 1620: $C_3^4:F_5$ x 2, $C_3^4:D_{10}$, $C_3^4:C_{20}$, $C_3^4:C_5:C_4$

Order 1458: $C_3^3\wr C_2$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.C_6$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1440: $F_9:F_5$

Order 1296: $C_3^4:\SD_{16}$ x 6, $C_3^2:(C_2\times F_9)$ x 6, $C_3^4:\SD_{16}$ x 4, $C_3^2\wr C_2.D_4$ x 4, $(C_3^2\times S_3^2):C_4$ x 3, $C_2\times C_3^4:D_4$ x 2, $C_3^4:(C_2\times Q_8)$ x 2, $C_3^4:(C_2\times D_4)$ x 2, $S_3^2:S_3^2$ x 2, $C_2\times C_3^4:Q_8$, $C_3^4:(C_2^2\times C_4)$, $C_2\times C_3^2:F_9$, $C_2\times C_3^2:F_9$, $C_3^2\wr C_2.Q_8$, $S_3\wr S_3$, $(C_3^3\times C_6).Q_8$, $(C_3^3\times C_6).D_4$, $(C_3^3\times C_6).D_4$, $C_3^3:C_{12}:C_4$, $C_3^2:C_6^2:C_4$

Order 1215: $C_3\wr C_5$

Order 972: $C_3^3:S_3^2$ x 10, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:C_3:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^4:D_6$, $C_3^3:S_3^2$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3.S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$, $C_3^3:D_{18}$

Order 960: $C_8:S_5$

Order 864: $F_9:D_6$

Order 810: $C_3^4:D_5$ x 2, $C_3^4:C_{10}$

Order 729: $C_3^6$, $C_3^5.C_3$, $C_3^5:C_3$

Order 720: $D_5\times F_9$, $D_5.\PSU(3,2)$, $D_5.\SOPlus(4,2)$

Order 648: $C_3^4:(C_2\times C_4)$ x 13, $C_3^4:D_4$ x 12, $C_3^3:(C_4\times S_3)$ x 11, $C_3^4:Q_8$ x 10, $C_3:S_3^3$ x 9, $C_3:S_3^3$ x 6, $C_3^2:F_9$ x 6, $C_3^2:F_9$ x 5, $C_2\times C_3^4:C_4$ x 4, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $C_3^4:C_2^3$ x 2, $C_3^3:(C_4\times S_3)$, $S_3\wr C_3$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:(C_4\times S_3)$, $C_3.S_3^3$, $C_2\times C_3^3:D_6$

Order 576: $F_9:D_4$

Order 486: $C_3^4:C_6$ x 10, $C_3^4:C_6$ x 10, $C_3^4:S_3$ x 5, $C_3^3:D_9$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^3:D_9$, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^3:C_{18}$, $C_3^4.S_3$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 480: $A_5:Q_8$, $C_4:S_5$, $C_8\times A_5$

Order 432: $S_3^3:C_2$ x 5, $S_3\times \PSU(3,2)$, $F_9:S_3$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $F_9:C_6$, $S_3\times F_9$

Order 405: $C_3^4:C_5$

Order 360: $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $D_5.S_3^2$, $C_5:F_9$, $C_5\times F_9$

Order 324: $C_3^2:S_3^2$ x 47, $C_3^4:C_4$ x 35, $C_3^2:S_3^2$ x 19, $C_3\wr C_2^2$ x 18, $C_3\wr C_4$ x 15, $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 9, $C_3^2\times S_3^2$ x 9, $C_3^3:D_6$ x 6, $C_3^2:C_6^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^2:D_{18}$, $C_3^3:D_6$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^3:A_4$, $C_3^3:C_{12}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2.S_3^2$, $C_3^3:D_6$

Order 288: $F_9:C_4$ x 3, $F_9:C_2^2$ x 3, $C_6^2:Q_8$, $C_6^2:D_4$, $\PSU(3,2):C_4$, $\SOPlus(4,2):C_4$, $C_2^2\times F_9$

Order 243: $C_3^5$ x 5, $C_3^4:C_3$ x 2, $C_3^3:C_9$ x 2, $C_3^4.C_3$, $C_3^4:C_3$

Order 240: $C_4\times A_5$, $A_5:C_4$, $C_2\times S_5$

Order 216: $S_3^3$ x 18, $C_3^2:C_4\times S_3$ x 11, $S_3^2:S_3$ x 9, $C_3^2:D_{12}$ x 5, $S_3^2:C_6$ x 5, $C_6:S_3^2$ x 4, $C_3^3:Q_8$ x 2, $C_6.S_3^2$ x 2, $C_3^3:Q_8$, $C_3:F_9$, $C_3\times F_9$, $C_6.S_3^2$

Order 192: $C_8:S_4$

Order 180: $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_{15}:C_{12}$

Order 162: $C_3^2\wr C_2$ x 79, $C_3^3:S_3$ x 54, $C_3^3:C_6$ x 19, $S_3\times C_3^3$ x 19, $C_3^3:C_6$ x 5, $C_3^3:S_3$ x 4, $C_3\wr S_3$ x 4, $C_3^2:D_9$ x 2, $C_3^3:C_6$ x 2, $C_3^3:C_6$ x 2, $C_3^3\times C_6$, $C_3^3:S_3$, $C_3^3.S_3$, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $D_9:C_3^2$, $S_3\times \He_3$, $C_3^3:C_6$, $C_3^2:D_9$

Order 160: $C_{40}:C_4$

Order 144: $F_9:C_2$ x 10, $C_2\times F_9$ x 8, $S_3^2:C_2^2$ x 5, $C_2.\PSU(3,2)$ x 4, $C_2\times \PSU(3,2)$ x 3, $C_2.\SOPlus(4,2)$ x 3, $S_3^2:C_4$ x 3, $C_6^2:C_4$ x 2, $C_6^2:C_4$, $D_6:D_6$, $(C_3\times C_{12}):C_4$

Order 120: $S_3\times F_5$, $C_2\times A_5$, $S_5$

Order 108: $C_3:S_3^2$ x 100, $C_3\times S_3^2$ x 44, $C_3:S_3^2$ x 18, $C_3^3:C_4$ x 11, $C_3^2:C_{12}$ x 11, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 4, $C_3^2\times D_6$ x 4, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$, $C_{18}:C_6$, $S_3\times D_9$

Order 96: $C_8\times A_4$, $C_4:S_4$, $A_4:Q_8$, $S_3\times \SD_{16}$

Order 90: $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 81: $C_3^4$ x 53, $C_3\wr C_3$ x 4, $C_3^2:C_9$ x 2, $C_9:C_3^2$, $C_3\times \He_3$

Order 80: $C_{20}:C_4$ x 2, $C_8\times D_5$

Order 72: $C_2\times C_3^2:C_4$ x 22, $\SOPlus(4,2)$ x 18, $S_3\times D_6$ x 16, $F_9$ x 11, $\PSU(3,2)$ x 10, $C_6.D_6$ x 7, $C_6:D_6$ x 2, $C_{12}:S_3$, $C_6\wr C_2$, $C_3:D_{12}$

Order 64: $C_8:D_4$

Order 60: $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$, $A_5$

Order 54: $C_3^2:S_3$ x 117, $C_3^2:C_6$ x 92, $S_3\times C_3^2$ x 92, $C_3^2:C_6$ x 6, $C_3^2\times C_6$ x 6, $C_3^2:S_3$ x 3, $C_9:C_6$ x 3, $C_3:D_9$, $S_3\times C_9$, $C_3\times D_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 2, $C_{24}:C_2$, $C_2\times S_4$, $S_3\times Q_8$, $S_3\times C_8$, $C_4\times A_4$, $A_4:C_4$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$, $C_{40}$, $C_5:C_8$

Order 36: $S_3^2$ x 70, $C_3^2:C_4$ x 42, $C_6:S_3$ x 30, $C_6\times S_3$ x 23, $C_3:C_{12}$ x 9, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $D_{18}$

Order 32: $D_4:C_4$, $C_2\times \SD_{16}$, $C_2^2\times C_8$, $C_2^2:Q_8$, $C_4:D_4$, $C_8:C_4$, $Q_8:C_4$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$ x 111, $C_9:C_3$ x 2, $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_4\times S_3$ x 5, $C_2\times D_6$ x 5, $C_3:D_4$ x 3, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2, $C_{24}$, $C_2\times A_4$, $S_4$, $C_3\times Q_8$, $C_3:C_8$

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3:S_3$ x 80, $C_3\times S_3$ x 73, $C_3\times C_6$ x 26, $D_9$ x 2, $C_{18}$

Order 16: $C_4:C_4$ x 3, $\SD_{16}$ x 2, $C_2\times C_8$ x 2, $C_2^2:C_4$ x 2, $C_2\times D_4$ x 2, $C_2\times Q_8$, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 25, $C_2\times C_6$ x 5, $C_{12}$ x 5, $C_3:C_4$ x 5, $A_4$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 54, $C_9$

Order 8: $C_2\times C_4$ x 5, $D_4$ x 4, $C_2^3$ x 2, $Q_8$ x 2, $C_8$ x 2

Order 6: $S_3$ x 19, $C_6$ x 13

Order 5: $C_5$

Order 4: $C_2^2$ x 6, $C_4$ x 5

Order 3: $C_3$ x 6

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 699840: $C_3^6:(C_8:S_5)$

Order 349920: $C_3^6.C_8.A_5$, $C_3^6.C_4.S_5$, $C_3^6:A_5:Q_8$

Order 174960: $C_3^6:(C_2\times S_5)$, $C_3^6:(C_4\times A_5)$, $C_3^6:A_5:C_4$

Order 139968: $C_3^6:(C_8:S_4)$

Order 116640: $C_3^6.C_{40}:C_4$

Order 87480: $C_3^6:S_5$, $C_3^6:(C_2\times A_5)$

Order 69984: $C_3^5:D_6:D_{12}$, $C_3^6:(S_3\times \SD_{16})$, $(C_3:S_3)^3.D_6$, $C_3^6:(C_8\times A_4)$

Order 58320: $C_3^6.C_{20}:C_4$, $C_3^4:(D_5\times F_9)$, $C_3^6:(C_{20}:C_4)$

Order 46656: $C_3^6:C_8:D_4$

Order 43740: $C_3^6:A_5$

Order 34992: $C_3^5:S_3.S_4$, $C_3^6.(C_2\times S_4)$, $C_3^6.(S_3\times D_4)$, $C_3^6.(C_4\times A_4)$, $C_3^6.(S_3\times Q_8)$, $C_3^2\wr C_3:\SD_{16}$, $C_3^6:(S_3\times C_8)$, $C_3^2\wr C_3:\SD_{16}$, $C_3^2\wr C_3:\SD_{16}$, $C_3^6:(C_3\times \SD_{16})$

Order 29160: $C_3^6:(C_2\times F_5)$, $C_3^5:(S_3\times F_5)$, $C_3^6:(C_4\times D_5)$, $C_3^5:C_{15}:C_8$, $C_3^6.C_{40}$

Order 23328: $C_3^6.(C_2\times \SD_{16})$, $C_3^6:(C_2^2\times C_8)$, $C_3^6.C_8:C_4$, $C_3^5:C_6.Q_{16}$, $C_3^6.C_4:D_4$, $C_3^5:C_6.D_8$, $C_3^4.C_6^2:Q_8$

Order 17496: $C_3^2\wr C_3:Q_8$ x 2, $C_3^6:(C_4\times S_3)$ x 2, $C_3^2\wr C_3:Q_8$, $C_3^2\wr C_3:C_8$, $C_3^6.C_{24}$, $C_3^6.S_4$, $C_3^2\wr C_3:D_4$, $C_3^6.D_{12}$, $C_3^6.(C_2\times A_4)$, $C_3^4.S_3^3$, $C_3^2\wr C_3:D_4$

Order 14580: $C_3^6:D_{10}$, $C_3^5:C_{15}:C_4$, $C_3^6:F_5$, $C_3^6:F_5$, $C_3^6.C_{20}$

Order 12960: $C_3^4:C_{40}:C_4$

Order 11664: $C_3^6.C_8.C_2$, $C_3^6.\SD_{16}$, $C_3^5:C_6.D_4$, $C_3^6:(C_2\times D_4)$, $C_3^5:C_6.D_4$, $C_3^6:(C_2^2\times C_4)$, $C_3^6.(C_2\times D_4)$, $C_3^6.C_4:C_4$, $C_3^6.\SD_{16}$, $C_3^5:C_{12}:C_4$, $C_3^6.(C_2\times Q_8)$, $C_3^4.(C_2\times F_9)$, $C_3^6.C_4:C_4$

Order 9720: $C_3^4:(S_3\times F_5)$

Order 8748: $C_3^2\wr C_3:C_4$ x 2, $C_3^6.C_{12}$ x 2, $C_3^6.D_6$, $C_3^6.A_4$, $C_3^6:D_6$, $C_3^6.D_6$, $C_3^2\wr C_3:C_2^2$, $C_3\wr D_6$

Order 7776: $C_3^4:(S_3\times \SD_{16})$

Order 7290: $C_3^6:C_{10}$, $C_3^6:D_5$, $C_3^5:D_{15}$

Order 6480: $C_3^4:C_{20}:C_4$ x 2, $C_3^4:(C_8\times D_5)$

Order 5832: $C_3^6:(C_2\times C_4)$ x 3, $C_3^6:Q_8$ x 2, $C_3^6:D_4$ x 2, $C_3^6.C_8$, $(C_3:S_3)^3$, $C_3^6:C_8$, $C_3^6:(C_2\times C_4)$, $C_3^6:(C_2\times C_4)$, $C_3^6:C_2^3$, $C_3^6:D_4$, $C_3^5:D_{12}$, $C_3^3.S_3^3$

Order 5184: $C_3^2:F_9:D_4$

Order 4860: $C_3\wr F_5$, $C_3^5:D_{10}$, $C_3^5:F_5$

Order 4374: $C_3^6:S_3$, $C_3^6.C_6$, $C_3^2\wr S_3$, $C_3\wr S_3$, $C_3\wr C_6$

Order 3888: $C_3^2:F_9:S_3$, $C_3^3:F_9:C_2$, $C_3^4:(C_3\times \SD_{16})$, $C_3^4:(S_3\times Q_8)$, $C_3^3:(C_2\times F_9)$, $C_3^3:F_9:C_2$, $S_3\times C_3^4:D_4$, $C_3^4:(S_3\times D_4)$

Order 3645: $C_3^5:C_{15}$

Order 3240: $C_3^4:(C_2\times F_5)$ x 2, $C_3^4:C_5:C_8$, $C_3^4:(C_4\times D_5)$, $C_3^4:C_{40}$

Order 2916: $C_3^6.C_4$ x 3, $C_3^5.D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^4:C_6^2$, $C_3^5:C_{12}$, $C_3^4:C_6^2$, $C_3^5:C_{12}$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:(C_2\times C_6)$, $C_3^5.D_6$, $C_3^5:D_6$

Order 2592: $C_3^2\wr C_2.\SD_{16}$ x 2, $C_3^4:(C_2\times \SD_{16})$ x 2, $(C_3^3\times C_6).Q_{16}$, $(C_3^3\times C_6).\SD_{16}$, $C_2\times C_3^4:\SD_{16}$, $C_3^2:(C_2^2\times F_9)$, $(C_3^3\times C_6).D_8$, $C_3^4:C_4:D_4$, $C_3^2:S_3^2:Q_8$

Order 2430: $C_3\wr D_5$, $C_3^4:D_{15}$, $C_3^5:C_{10}$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^5:D_4$ x 2, $C_3\wr C_4:S_3$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^4:C_4\times S_3$, $C_3^2.S_3^3$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^4:D_{12}$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$, $C_3^4:C_{24}$

Order 1620: $C_3^4:F_5$ x 2, $C_3^4:D_{10}$, $C_3^4:C_{20}$, $C_3^4:C_5:C_4$

Order 1458: $C_3^3\wr C_2$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.C_6$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1440: $F_9:F_5$

Order 1296: $C_3^4:\SD_{16}$ x 6, $C_3^2:(C_2\times F_9)$ x 6, $C_3^4:\SD_{16}$ x 4, $C_3^2\wr C_2.D_4$ x 4, $(C_3^2\times S_3^2):C_4$ x 3, $C_2\times C_3^4:D_4$ x 2, $C_3^4:(C_2\times Q_8)$ x 2, $C_3^4:(C_2\times D_4)$ x 2, $S_3^2:S_3^2$ x 2, $C_2\times C_3^4:Q_8$, $C_3^4:(C_2^2\times C_4)$, $C_2\times C_3^2:F_9$, $C_2\times C_3^2:F_9$, $C_3^2\wr C_2.Q_8$, $S_3\wr S_3$, $(C_3^3\times C_6).Q_8$, $(C_3^3\times C_6).D_4$, $(C_3^3\times C_6).D_4$, $C_3^3:C_{12}:C_4$, $C_3^2:C_6^2:C_4$

Order 1215: $C_3\wr C_5$

Order 972: $C_3^3:S_3^2$ x 10, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:C_3:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^4:D_6$, $C_3^3:S_3^2$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3.S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$, $C_3^3:D_{18}$

Order 960: $C_8:S_5$

Order 864: $F_9:D_6$

Order 810: $C_3^4:D_5$ x 2, $C_3^4:C_{10}$

Order 729: $C_3^6$, $C_3^5.C_3$, $C_3^5:C_3$

Order 720: $D_5\times F_9$, $D_5.\PSU(3,2)$, $D_5.\SOPlus(4,2)$

Order 648: $C_3^4:(C_2\times C_4)$ x 13, $C_3^4:D_4$ x 12, $C_3^3:(C_4\times S_3)$ x 11, $C_3^4:Q_8$ x 10, $C_3:S_3^3$ x 9, $C_3:S_3^3$ x 6, $C_3^2:F_9$ x 6, $C_3^2:F_9$ x 5, $C_2\times C_3^4:C_4$ x 4, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $C_3^4:C_2^3$ x 2, $C_3^3:(C_4\times S_3)$, $S_3\wr C_3$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:(C_4\times S_3)$, $C_3.S_3^3$, $C_2\times C_3^3:D_6$

Order 576: $F_9:D_4$

Order 486: $C_3^4:C_6$ x 10, $C_3^4:C_6$ x 10, $C_3^4:S_3$ x 5, $C_3^3:D_9$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^3:D_9$, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^3:C_{18}$, $C_3^4.S_3$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 480: $A_5:Q_8$, $C_4:S_5$, $C_8\times A_5$

Order 432: $S_3^3:C_2$ x 5, $S_3\times \PSU(3,2)$, $F_9:S_3$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $F_9:C_6$, $S_3\times F_9$

Order 405: $C_3^4:C_5$

Order 360: $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $D_5.S_3^2$, $C_5:F_9$, $C_5\times F_9$

Order 324: $C_3^2:S_3^2$ x 47, $C_3^4:C_4$ x 35, $C_3^2:S_3^2$ x 19, $C_3\wr C_2^2$ x 18, $C_3\wr C_4$ x 15, $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 9, $C_3^2\times S_3^2$ x 9, $C_3^3:D_6$ x 6, $C_3^2:C_6^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^2:D_{18}$, $C_3^3:D_6$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^3:A_4$, $C_3^3:C_{12}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2.S_3^2$, $C_3^3:D_6$

Order 288: $F_9:C_4$ x 3, $F_9:C_2^2$ x 3, $C_6^2:Q_8$, $C_6^2:D_4$, $\PSU(3,2):C_4$, $\SOPlus(4,2):C_4$, $C_2^2\times F_9$

Order 243: $C_3^5$ x 5, $C_3^4:C_3$ x 2, $C_3^3:C_9$ x 2, $C_3^4.C_3$, $C_3^4:C_3$

Order 240: $C_4\times A_5$, $A_5:C_4$, $C_2\times S_5$

Order 216: $S_3^3$ x 18, $C_3^2:C_4\times S_3$ x 11, $S_3^2:S_3$ x 9, $C_3^2:D_{12}$ x 5, $S_3^2:C_6$ x 5, $C_6:S_3^2$ x 4, $C_3^3:Q_8$ x 2, $C_6.S_3^2$ x 2, $C_3^3:Q_8$, $C_3:F_9$, $C_3\times F_9$, $C_6.S_3^2$

Order 192: $C_8:S_4$

Order 180: $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_{15}:C_{12}$

Order 162: $C_3^2\wr C_2$ x 79, $C_3^3:S_3$ x 54, $C_3^3:C_6$ x 19, $S_3\times C_3^3$ x 19, $C_3^3:C_6$ x 5, $C_3^3:S_3$ x 4, $C_3\wr S_3$ x 4, $C_3^2:D_9$ x 2, $C_3^3:C_6$ x 2, $C_3^3:C_6$ x 2, $C_3^3\times C_6$, $C_3^3:S_3$, $C_3^3.S_3$, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $D_9:C_3^2$, $S_3\times \He_3$, $C_3^3:C_6$, $C_3^2:D_9$

Order 160: $C_{40}:C_4$

Order 144: $F_9:C_2$ x 10, $C_2\times F_9$ x 8, $S_3^2:C_2^2$ x 5, $C_2.\PSU(3,2)$ x 4, $C_2\times \PSU(3,2)$ x 3, $C_2.\SOPlus(4,2)$ x 3, $S_3^2:C_4$ x 3, $C_6^2:C_4$ x 2, $C_6^2:C_4$, $D_6:D_6$, $(C_3\times C_{12}):C_4$

Order 120: $S_3\times F_5$, $C_2\times A_5$, $S_5$

Order 108: $C_3:S_3^2$ x 100, $C_3\times S_3^2$ x 44, $C_3:S_3^2$ x 18, $C_3^3:C_4$ x 11, $C_3^2:C_{12}$ x 11, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 4, $C_3^2\times D_6$ x 4, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$, $C_{18}:C_6$, $S_3\times D_9$

Order 96: $C_8\times A_4$, $C_4:S_4$, $A_4:Q_8$, $S_3\times \SD_{16}$

Order 90: $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 81: $C_3^4$ x 53, $C_3\wr C_3$ x 4, $C_3^2:C_9$ x 2, $C_9:C_3^2$, $C_3\times \He_3$

Order 80: $C_{20}:C_4$ x 2, $C_8\times D_5$

Order 72: $C_2\times C_3^2:C_4$ x 22, $\SOPlus(4,2)$ x 18, $S_3\times D_6$ x 16, $F_9$ x 11, $\PSU(3,2)$ x 10, $C_6.D_6$ x 7, $C_6:D_6$ x 2, $C_{12}:S_3$, $C_6\wr C_2$, $C_3:D_{12}$

Order 64: $C_8:D_4$

Order 60: $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$, $A_5$

Order 54: $C_3^2:S_3$ x 117, $C_3^2:C_6$ x 92, $S_3\times C_3^2$ x 92, $C_3^2:C_6$ x 6, $C_3^2\times C_6$ x 6, $C_3^2:S_3$ x 3, $C_9:C_6$ x 3, $C_3:D_9$, $S_3\times C_9$, $C_3\times D_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 2, $C_{24}:C_2$, $C_2\times S_4$, $S_3\times Q_8$, $S_3\times C_8$, $C_4\times A_4$, $A_4:C_4$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$, $C_{40}$, $C_5:C_8$

Order 36: $S_3^2$ x 70, $C_3^2:C_4$ x 42, $C_6:S_3$ x 30, $C_6\times S_3$ x 23, $C_3:C_{12}$ x 9, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $D_{18}$

Order 32: $D_4:C_4$, $C_2\times \SD_{16}$, $C_2^2\times C_8$, $C_2^2:Q_8$, $C_4:D_4$, $C_8:C_4$, $Q_8:C_4$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$ x 111, $C_9:C_3$ x 2, $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_4\times S_3$ x 5, $C_2\times D_6$ x 5, $C_3:D_4$ x 3, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2, $C_{24}$, $C_2\times A_4$, $S_4$, $C_3\times Q_8$, $C_3:C_8$

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3:S_3$ x 80, $C_3\times S_3$ x 73, $C_3\times C_6$ x 26, $D_9$ x 2, $C_{18}$

Order 16: $C_4:C_4$ x 3, $\SD_{16}$ x 2, $C_2\times C_8$ x 2, $C_2^2:C_4$ x 2, $C_2\times D_4$ x 2, $C_2\times Q_8$, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 25, $C_2\times C_6$ x 5, $C_{12}$ x 5, $C_3:C_4$ x 5, $A_4$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 54, $C_9$

Order 8: $C_2\times C_4$ x 5, $D_4$ x 4, $C_2^3$ x 2, $Q_8$ x 2, $C_8$ x 2

Order 6: $S_3$ x 19, $C_6$ x 13

Order 5: $C_5$

Order 4: $C_2^2$ x 6, $C_4$ x 5

Order 3: $C_3$ x 6

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 699840: $C_3^6:(C_8:S_5)$ ($C_1$)

Order 349920: $C_3^6.C_8.A_5$ ($C_2$), $C_3^6.C_4.S_5$ ($C_2$), $C_3^6:A_5:Q_8$ ($C_2$)

Order 174960: $C_3^6:(C_4\times A_5)$ ($C_2^2$)

Order 87480: $C_3^6:(C_2\times A_5)$ ($D_4$)

Order 43740: $C_3^6:A_5$ ($\SD_{16}$)

Order 5832: $C_3^6.C_8$ ($S_5$)

Order 2916: $C_3^6.C_4$ ($C_2\times S_5$)

Order 1458: $C_3^5:S_3$ ($C_4:S_5$)

Order 729: $C_3^6$ ($C_8:S_5$)

Order 1: $C_1$ ($C_3^6:(C_8:S_5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 699840: $C_3^6:(C_8:S_5)$ ($C_1$)

Order 349920: $C_3^6.C_8.A_5$ ($C_2$), $C_3^6.C_4.S_5$ ($C_2$), $C_3^6:A_5:Q_8$ ($C_2$)

Order 174960: $C_3^6:(C_4\times A_5)$ ($C_2^2$)

Order 87480: $C_3^6:(C_2\times A_5)$ ($D_4$)

Order 43740: $C_3^6:A_5$ ($\SD_{16}$)

Order 5832: $C_3^6.C_8$ ($S_5$)

Order 2916: $C_3^6.C_4$ ($C_2\times S_5$)

Order 1458: $C_3^5:S_3$ ($C_4:S_5$)

Order 729: $C_3^6$ ($C_8:S_5$)

Order 1: $C_1$ ($C_3^6:(C_8:S_5)$)

Series

Derived series

$C_3^6:(C_8:S_5)$

$\rhd$

$C_3^6:(C_8:S_5)$

$\rhd$

$C_3^6:(C_4\times A_5)$

$\rhd$

$C_3^6:(C_4\times A_5)$

$\rhd$

$C_3^6:A_5$

$\rhd$

$C_3^6:A_5$

Chief series

$C_3^6:(C_8:S_5)$

$\rhd$

$C_3^6:(C_8:S_5)$

$\rhd$

$C_3^6.C_8.A_5$

$\rhd$

$C_3^6.C_8.A_5$

$\rhd$

$C_3^6.C_8$

$\rhd$

$C_3^6.C_8$

$\rhd$

$C_3^6.C_4$

$\rhd$

$C_3^6.C_4$

$\rhd$

$C_3^5:S_3$

$\rhd$

$C_3^5:S_3$

$\rhd$

$C_3^6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^6:(C_8:S_5)$

$\rhd$

$C_3^6:(C_8:S_5)$

$\rhd$

$C_3^6:(C_4\times A_5)$

$\rhd$

$C_3^6:(C_4\times A_5)$

$\rhd$

$C_3^6:(C_2\times A_5)$

$\rhd$

$C_3^6:(C_2\times A_5)$

$\rhd$

$C_3^6:A_5$

$\rhd$

$C_3^6:A_5$

Upper central series

$C_1$

$\lhd$

$C_1$

Character theory

Complex character table

See the $53 \times 53$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $44 \times 44$ rational character table.