Abstract group 559872000.f: $A_6^3.D_6$

• Label: 559872000.f
• Order: \( 2^{11} \cdot 3^{7} \cdot 5^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{7} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $30$
• Trans deg.: $30$
• Rank: $2$

Group information

Description:	$A_6^3.D_6$
Order:	\(559872000\)\(\medspace = 2^{11} \cdot 3^{7} \cdot 5^{3} \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$A_6\wr C_3.C_2^3$, of order \(1119744000\)\(\medspace = 2^{12} \cdot 3^{7} \cdot 5^{3} \)
Composition factors:	$C_2$ x 2, $C_3$, $A_6$ x 3
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	20	24	30	40	60
Elements	1	147015	790640	23899320	3048624	24008400	97977600	20736000	33417360	102859200	45135360	61663680	69984000	27993600	41990400	6220800	559872000
Conjugacy classes	1	5	6	13	5	10	19	1	9	10	9	8	5	6	3	2	112
Divisions	1	5	6	13	5	10	12	1	9	10	6	8	3	3	2	1	95
Autjugacy classes	1	5	6	13	5	10	12	1	9	10	6	8	3	3	2	1	95

Dimension	1	2	27	30	48	60	150	243	250	300	384	480	500	540	600	729	750	864	960	1000	1024	1080	1200	1350	1458	1500	1920	2000	2048	2400	2430	2700	3000	3072	3456	3840	3888	4000	4320	4800	4860	5400	7680	8640	9600
Irr. complex chars.	4	2	4	6	2	0	4	4	2	4	4	2	1	3	1	4	2	1	1	4	2	0	4	4	2	4	4	2	1	1	6	4	2	2	4	5	2	0	2	4	0	1	0	1	0	112
Irr. rational chars.	4	2	4	2	2	2	4	4	2	4	4	0	1	1	1	4	2	1	2	0	2	1	0	4	2	2	0	2	1	3	2	4	3	2	4	5	2	1	0	2	2	1	1	2	1	95

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$30$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	27	27	27
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $30$ $\langle(1,17,10,20,2,15,9,19)(3,18)(4,13,5,11,7,12,8,16)(6,14)(22,29,24,26)(23,30,27,28) \!\cdots\! \rangle$
Transitive group:	30T4733	36T88326			more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(A_6\wr S_3)$ . $C_2$	$(A_6^3.S_3)$ . $C_2$	$(A_6^3.C_6)$ . $C_2$	$(A_6.A_6.A_6)$ . $D_6$	all 6

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 8 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $A_6\wr C_3$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 41472 (order at least 13500) are shown, as well as all normal subgroups of any index.

Order 559872000: $A_6^3.D_6$

Order 279936000: $A_6^3.C_6$, $A_6^3.S_3$, $A_6\wr S_3$

Order 186624000: $A_6.A_6^2.C_2^2$

Order 139968000: $A_6\wr C_3$

Order 93312000: $A_6.A_6^2.C_2$ x 2, $A_6.A_6.A_6.C_2$

Order 46656000: $A_6.A_6.A_6$

Order 18662400: $C_3:S_3.C_2.A_6^2.C_2^2$

Order 15552000: $A_5.A_6^2.C_2$

Order 9331200: $C_3:S_3.A_6^2.C_2^2$ x 4, $C_3:S_3.C_2.A_6^2.C_2$ x 2, $C_3:S_3.C_2.A_6.A_6.C_2$

Order 7776000: $A_5\times A_6^2$

Order 6220800: $S_4.A_6^2.C_2$

Order 5184000: $D_5.A_6^2.C_2^2$

Order 4665600: $C_3^2.A_6^2:C_4$ x 3, $C_3:S_3.A_6.A_6.C_2$ x 2, $C_3:S_3.C_2\times A_6^2$, $C_3:S_3.A_6^2.C_2$

Order 4147200: $D_4.A_6^2.C_2^2$

Order 3110400: $A_4.A_6^2.C_2$ x 2, $S_4\times A_6^2$

Order 2592000: $D_5.A_6^2.C_2$ x 2, $A_6.A_5^2.C_2$ x 2, $D_5.A_6.A_6.C_2$

Order 2332800: $C_3^2.A_6^2.C_2$ x 2, $C_3:S_3.A_6.A_6$

Order 2073600: $C_4.A_6^2.C_2^2$ x 4, $D_4.A_6^2.C_2$ x 2, $D_4.A_6.A_6.C_2$

Order 1866240: $C_3^4.C_4\wr C_2.A_6.C_2$

Order 1555200: $A_4.A_6.A_6$, $A_6^2.D_6$

Order 1296000: $C_5.A_6^2.C_2$ x 2, $D_5.A_6.A_6$, $A_5.A_6.A_5$, $A_5.A_5.A_6$, $A_5\wr S_3$

Order 1166400: $C_3^2.A_6.A_6$

Order 1036800: $C_4.A_6^2.C_2$ x 4, $C_4.A_6.A_6.C_2$ x 2, $C_2.A_6^2.C_2^2$ x 2, $D_4.A_6.A_6$, $C_2^2.A_6^2.C_2$, $A_6^2.D_4$

Order 933120: $C_3^4.\OD_{16}.A_6.C_2$ x 2, $C_3^4.D_4:C_2.A_6.C_2$ x 2, $C_3^4.C_4^2.A_6.C_2$ x 2, $C_3^4.C_4\wr C_2.A_6$

Order 777600: $C_3:S_3.C_2.A_6.A_5$, $S_3\times A_6^2$, $A_6^2:C_6$, $A_6^2:S_3$

Order 648000: $C_5.A_6.A_6$, $A_5\wr C_3$

Order 559872: $C_3^6:(C_4^3.D_6)$

Order 518400: $A_6.A_6.C_2^2$ x 3, $C_2.A_6.A_6.C_2$ x 2, $S_4\times A_5\times A_6$ x 2, $A_6^2:C_4$ x 2, $A_6^2:C_2^2$ x 2, $C_4.A_6.A_6$, $C_2^2.A_6.A_6$

Order 466560: $C_3^4:C_2^2.C_2.A_6.C_2$ x 4, $C_3^4.D_4.A_6.C_2$ x 4, $C_3^4.(C_2\times C_4).A_6.C_2$ x 4, $C_3^4.Q_8.A_6.C_2$ x 2, $C_3^4.C_8.A_6.C_2$ x 2, $C_3^4.\OD_{16}.A_6$, $C_3^4.D_4:C_2.A_6$, $C_3^4.C_4^2.A_6$

Order 432000: $A_5^3:C_2$ x 2

Order 414720: $A_4^2.C_2^2.A_6.C_2$, $A_6\times A_4^2.D_4$

Order 388800: $C_3:S_3.A_5.A_6$, $C_3\times A_6^2$

Order 311040: $C_6^2.(C_4\times S_3).A_6$

Order 279936: $C_3^5:D_6.(C_4\times S_4)$ x 2, $(C_3:S_3)^3.(C_2\times S_4)$ x 2, $C_3^6:C_4\wr S_3$, $C_3^6:(C_4^3.S_3)$, $C_3^6:(C_4^3.C_6)$

Order 259200: $A_6.A_6.C_2$ x 10, $C_3:S_3.C_2.A_5^2.C_2$ x 2, $A_4\times A_5\times A_6$ x 2, $C_2\times A_6^2$ x 2, $A_6\wr C_2$ x 2, $(D_5\times C_3:S_3.C_2).A_6.C_2$, $A_6\times S_6$

Order 233280: $C_3^2:(C_3:S_3.C_2).A_6.C_2$ x 12, $C_3^4:C_2^2.A_6.C_2$ x 8, $C_3^4.D_4.A_6$ x 2, $C_3^4.(C_2\times C_4).A_6$ x 2, $C_3^4:C_2^2.C_2.A_6$, $C_3^4.Q_8.A_6$, $C_3^4.C_8.A_6$

Order 216000: $A_5^3$ x 2, $A_5\times A_6\times D_5$

Order 207360: $\PSOPlusPlus(4,3).A_6.C_2$ x 2, $A_6.S_4^2$ x 2, $(D_4\times C_3:S_3.C_2).A_6.C_2$, $A_4^2:C_4\times A_6$, $A_6\times \POPlus(4,3)$

Order 194400: $C_3^2.A_6.A_5$

Order 186624: $C_3^6.C_4^3.C_2^2$

Order 172800: $S_4\times \SOPlus(4,4)$ x 2, $D_4\times A_5\times A_6$

Order 155520: $C_3^4.C_4\wr C_2.A_5$, $(C_3^2\times A_4):C_4.A_6$, $(C_3:S_3\times S_4).A_6$, $(A_4\times C_3:S_3.C_2).A_6$

Order 144000: $D_5\wr C_2.A_6.C_2$

Order 139968: $C_3^6:(C_4^2:C_{12})$ x 2, $(C_3:S_3)^3.S_4$ x 2, $(C_3:S_3)^3.S_4$, $(C_3:S_3)^3.S_4$, $C_3^6:C_4\wr C_3$, $C_3^6:(Q_8\times S_4)$

Order 129600: $A_6^2$ x 5, $C_3^2:(A_5^2:C_4)$ x 3, $C_3:S_3.C_2.A_5.A_5$ x 2, $(D_5\times C_3:S_3).A_6.C_2$ x 2, $S_3\times A_5\times A_6$ x 2, $C_3:S_3.A_5^2.C_2$, $(D_5\times C_3:S_3.C_2).A_6$

Order 116640: $C_3:(C_3^3:C_2).A_6.C_2$ x 5, $C_3^2:(C_3:S_3.C_2).A_6$ x 3, $C_3^4:C_2^2.A_6$ x 2, $C_3^3:(C_3:C_4).A_6$, $(C_3^2\times C_3:S_3.C_2).A_6$

Order 108000: $C_5.A_6.A_5$

Order 103680: $(D_4\times C_3:S_3).A_6.C_2$ x 2, $(C_4\times C_3:S_3.C_2).A_6.C_2$ x 2, $(C_3\times C_{12}):C_4.A_6.C_2$ x 2, $A_4\times S_4\times A_6$ x 2, $A_6\times \PSOPlus(4,3)$ x 2, $A_4^2.A_6.C_2$, $(D_4\times C_3:S_3.C_2).A_6$, $A_4\wr C_2\times A_6$

Order 93312: $C_3^6.C_2^3.C_2^3.C_2$ x 5, $C_3^6.C_2.C_2^3.C_2^3$ x 4, $C_3^6.C_4^3.C_2$ x 3, $C_3^6.C_4^2.C_2^3$ x 2, $C_3^6.C_2^2.C_2^5$

Order 92160: $D_4^2.C_2.A_6.C_2$

Order 86400: $S_4\times A_5^2$ x 3, $A_5^2:S_4$ x 2, $A_4\times \SOPlus(4,4)$ x 2, $C_2^2\times A_5\times A_6$ x 2, $C_4\times A_5\times A_6$, $D_5\times S_4\times A_6$

Order 82944: $A_4^3.(C_2\times S_4)$

Order 77760: $C_3^4.\OD_{16}.A_5$, $C_3^4.D_4:C_2.A_5$, $C_3^4.C_4^2.A_5$, $C_3^2:S_4.A_6$, $(C_3^2\times S_4).A_6$, $(A_4\times C_3:S_3).A_6$, $C_3^2:C_4\times S_3\times A_6$

Order 72000: $D_5^2.A_6.C_2$ x 4, $C_5:F_5.A_6.C_2$ x 2, $D_5\wr C_2.A_6$, $A_5^2:D_{10}$, $A_5^2:F_5$

Order 69984: $C_3^6:(C_4\times S_4)$ x 3, $(C_3:S_3)^3.D_6$ x 3, $C_3^5:D_6.S_4$, $C_3^5:D_6.S_4$, $C_3^6:(C_4^2:C_6)$, $C_3^6:(Q_8\times A_4)$

Order 69120: $A_5\times S_4\wr C_2$ x 2, $C_2^2:S_4.A_6.C_2$, $D_4\times S_4\times A_6$, $A_6\times C_2^2:S_4:C_2$

Order 64800: $C_3^2.A_5^2.C_2$ x 2, $C_3:S_3.A_5.A_5$ x 2, $C_3:D_{15}.A_6.C_2$ x 2, $A_6\times \GL(2,4)$ x 2, $C_3^2:(C_5:C_4).A_6$, $(D_5\times C_3:S_3).A_6$, $(C_5\times C_3:S_3.C_2).A_6$

Order 62208: $C_3^4.A_4.C_4^2.C_2^2$

Order 58320: $(C_3^2\times C_3:S_3).A_6$ x 2, $C_3^4.A_6.C_2$, $C_3:(C_3^3:C_2).A_6$

Order 57600: $(D_4\times D_5).A_6.C_2$, $D_4\times \SOPlus(4,4)$, $A_5^2:\SD_{16}$

Order 51840: $(C_4\times C_3:S_3).A_6.C_2$ x 4, $(C_2\times C_3:S_3.C_2).A_6.C_2$ x 4, $\PSU(3,2).A_6.C_2$ x 2, $C_3:S_3.C_2.A_6.C_2^2$ x 2, $C_3^2:C_4\times S_4\times A_5$ x 2, $A_4^2\times A_6$ x 2, $S_3\times S_4\times A_6$ x 2, $S_3\wr C_2.A_6.C_2$, $F_9.A_6.C_2$, $C_3^4.C_{10}.C_2^3.C_2^3$, $C_3:S_3.D_4.A_6$, $(D_4\times C_3:S_3).A_6$, $(C_4\times C_3:S_3.C_2).A_6$, $(C_3\times C_{12}):C_4.A_6$, $(C_2^2\times C_3:S_3.C_2).A_6$

Order 46656: $C_3^6.C_2^3.C_2^3$ x 14, $C_3^6.C_4.C_2.C_2^3$ x 9, $C_3^6.C_4:Q_8.C_2$ x 6, $C_3^6.C_4.Q_8.C_2$ x 5, $C_3^6.\OD_{16}:C_4$ x 4, $C_3^6.C_4.C_4^2$ x 3, $C_3^6:(D_4\times Q_8)$ x 3, $C_3^4:(C_{12}^2:C_4)$ x 3, $C_3^6.Q_8^2$, $C_3^6.Q_8:Q_8$, $C_3^6.D_4:Q_8$, $C_3^6.C_8:Q_8$, $C_3^6.C_4^3$, $C_3^6.C_4^2.C_2^2$, $C_3^6.C_4:\OD_{16}$, $C_3^6.(C_4\times \OD_{16})$

Order 46080: $D_4^2.A_6.C_2$ x 2, $D_4:D_4.A_6.C_2$ x 2, $C_4^2:C_4.A_6.C_2$ x 2, $A_6\times C_2\wr D_4$

Order 43200: $A_4\times A_5^2$ x 3, $C_2\times A_5\times A_6$ x 3, $A_5^2:D_6$ x 2, $A_5\times A_6.C_2$ x 2, $A_5\times S_6$, $S_5\times A_6$, $A_5\times \PGL(2,9)$, $D_5\times A_4\times A_6$, $C_5:S_4\times A_6$, $C_5\times S_4\times A_6$, $A_6.S_5$

Order 41472: $C_3^4.C_2^3.D_4^2$, $A_4^2.C_3^2.C_2^2:Q_8$, $C_3^2:C_4\times S_4\wr C_2$, $S_4\wr C_3$, $A_4^3:S_4$, $A_4^3:S_4$

Order 38880: $C_3^4.D_4.A_5$ x 2, $C_3^4.(C_2\times C_4).A_5$ x 2, $C_3:S_3^2\times A_6$ x 2, $C_3^4:C_2^2.C_2.A_5$, $C_3^4.Q_8.A_5$, $C_3^4.C_8.A_5$, $C_3^3:C_2.A_6.C_2$, $(C_3^2\times A_4).A_6$, $C_3^3:C_4\times A_6$, $C_3^2:C_{12}\times A_6$

Order 36000: $C_5:D_5.A_6.C_2$ x 4, $D_5^2.A_6$ x 2, $D_5\times A_5^2$ x 2, $C_5:F_5.A_6$, $A_5^2:D_5$, $A_5^2:C_{10}$

Order 34992: $C_3^5:S_3.S_4$ x 3, $C_3^6.(C_4\times A_4)$ x 3, $C_3^6.(S_3\times Q_8)$, $C_3^6.(C_2\times S_4)$, $C_3^5:D_6.A_4$

Order 34560: $C_2^2:S_4\times A_6$ x 3, $A_5\times S_4^2$ x 3, $C_2^4:C_3.A_6.C_2$ x 2, $A_5\times \POPlus(4,3)$ x 2, $A_4^2:C_4\times A_5$ x 2, $A_6\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^2\times S_4\times A_6$ x 2, $C_2^3:A_4\times A_6$, $D_4\times A_4\times A_6$, $C_4\times S_4\times A_6$, $C_4:S_4\times A_6$

Order 32400: $C_3^2.A_5.A_5$ x 2, $C_3:D_{15}.A_6$, $(C_5\times C_3:S_3).A_6$, $(C_3^2\times D_5).A_6$

Order 31104: $C_3^4.A_4.C_4\wr C_2$ x 3, $C_6^2.C_3^3.C_2^3.C_2^2$, $C_3^4.A_4.C_4.C_2^3$, $C_3^4.A_4.C_2.C_2^4$, $C_3^4.(C_4^2\times A_4).C_2$

Order 29160: $C_3^4.A_6$

Order 28800: $(C_4\times D_5).A_6.C_2$ x 2, $A_5^2:D_4$ x 2, $D_4\times A_5^2$ x 2, $A_5^2:C_2^3$ x 2, $D_4\times D_5\times A_6$, $A_5^2:C_8$, $C_4\times \SOPlus(4,4)$, $A_5^2:Q_8$, $A_5^2:D_4$

Order 27648: $S_4^3:C_2$ x 2

Order 25920: $C_3:S_3.C_2.A_6.C_2$ x 30, $C_3:S_3.A_6.C_2^2$ x 6, $(C_2\times C_3:S_3).A_6.C_2$ x 5, $C_2\times C_3^2:C_4\times A_6$ x 5, $C_3:S_4\times A_6$ x 4, $A_6.\SOPlus(4,2)$ x 3, $C_3^2:C_4.A_6.C_2$ x 2, $A_6\times \SOPlus(4,2)$ x 2, $C_6^2:\GL(2,4):C_4$ x 2, $C_6^2:C_{12}\times A_5$ x 2, $S_4\times \GL(2,4):S_3$ x 2, $C_3\times S_4\times A_6$ x 2, $S_3\times A_4\times A_6$ x 2, $A_6\times \PSU(3,2)$ x 2, $(C_3^2\times S_6):C_4$ x 2, $C_6^2:C_2.A_6$, $C_3^4.\OD_{16}:F_5$, $C_3^4.Q_8:(C_2\times F_5)$, $C_3^4.D_{10}.C_2^4$, $C_3^4.C_{20}.C_4^2$, $C_3^4.C_{20}.C_4.C_2^2$, $C_3^4.C_4:(C_4:F_5)$, $C_3^4.(D_5.D_8):C_2$, $C_3:D_{12}.A_6$, $C_2\times C_3:S_3.A_6.C_2$, $(C_4\times C_3:S_3).A_6$, $(C_3^2\times D_4).A_6$, $(C_2^2\times C_3:S_3).A_6$, $C_3^2:C_4\times S_6$, $F_9\times A_6$, $A_6:\PSU(3,2)$

Order 23328: $C_3^4.C_6^2:Q_8$ x 15, $C_3^6.C_4:Q_8$ x 14, $C_3^6.C_4^2.C_2$ x 10, $C_3^6.C_4:Q_8$ x 8, $C_3^6.(C_4.Q_8)$ x 8, $C_3^6:(C_4\times D_4)$ x 8, $C_3^6.C_4\wr C_2$ x 7, $C_3^6.C_2^3.C_2^2$ x 7, $C_3^6.C_8:C_4$ x 6, $C_3^6.(C_2.D_8)$ x 6, $C_3^6:(C_4^2:C_2)$ x 6, $C_3^6.C_4:C_4.C_2$ x 4, $C_3^6.(C_4\times Q_8)$ x 4, $C_3^6.(C_2^2\times Q_8)$ x 4, $C_3^6.C_2^2:Q_8$ x 3, $C_3^6:(C_4\times Q_8)$ x 3, $C_3^6:(C_2\times \OD_{16})$ x 3, $C_3^6.Q_8:C_4$ x 2, $C_3^6.C_4:C_8$ x 2, $C_3^6.C_2^2.C_2^3$ x 2, $C_3^5:S_3.D_8$ x 2, $C_3^6.C_4.C_2^3$, $C_3^6.(C_4\times D_4)$, $C_3^5.D_6.C_2^3$, $C_3^4.C_{12}^2.C_2$

Order 23040: $C_4:D_4.A_6.C_2$ x 4, $C_8:C_2^2.A_6.C_2$ x 2, $C_4\wr C_2.A_6.C_2$ x 2, $C_2^2\wr C_2.A_6.C_2$ x 2, $(C_4\times D_4).A_6.C_2$ x 2, $Q_8:C_2^2.A_6.C_2$, $C_2^3.C_4.A_6.C_2$, $D_4^2\times A_6$, $D_4:D_4\times A_6$, $C_4^2:C_4\times A_6$, $C_2\wr C_2^2\times A_6$, $C_2\wr C_4\times A_6$

Order 21600: $A_5\times A_6$ x 9, $S_3\times A_5^2$ x 3, $A_5^2:S_3$ x 2, $A_5^2:C_6$ x 2, $(D_5\times C_3:S_3.C_2).A_5$, $S_3\times D_5\times A_6$, $C_5\times A_4\times A_6$

Order 20736: $C_3^4.C_4^3.C_2^2$ x 5, $(C_3^2\times S_4^2):C_4$ x 3, $C_3^4.C_4^2.C_4.C_2^2$ x 2, $C_3^4.C_4^2.C_2^4$ x 2, $A_4^2.C_3^2.C_4:C_4$ x 2, $C_3^2:C_4\times S_4^2$ x 2, $C_3^4.C_4^2.C_2.C_2^3$, $C_3^4.C_2^3.C_4:D_4$, $C_3^4.C_2^3.C_2^4.C_2$, $C_3^4.C_2^3.C_2^2:D_4$, $C_3^4.C_2^3.(C_4\times D_4)$, $C_3^4.(C_2\times D_4).C_2^4$, $C_3^2.A_4^2.C_4:C_4$, $C_3^2.A_4^2.C_2.C_2^3$, $A_4^3:A_4$, $A_4^3:D_6$, $A_4^2:C_6^2:C_4$, $(C_3\times A_4^2:C_{12}):C_4$, $A_4^2:C_6^2:C_4$, $(C_3^2\times A_4^2):C_4^2$, $(C_3\times S_4^2):D_6$

Order 19440: $C_3^4.(C_4\times A_5)$ x 5, $C_3^2:S_3\times A_6$ x 3, $C_3^4:C_2^2.A_5$ x 2, $C_3^3.A_6.C_2$ x 2, $S_3\times C_3^2\times A_6$ x 2, $C_3^2:C_6\times A_6$ x 2

Order 18000: $(C_5\times D_5).A_6$ x 2, $C_5\times A_5^2$ x 2, $C_5^2.A_6.C_2$, $C_5:D_5.A_6$

Order 17496: $C_3^2\wr C_3:Q_8$ x 3, $C_3^6:(C_4\times S_3)$ x 3, $C_3^6:S_4$, $C_3^2\wr C_3:Q_8$, $C_3^6:S_4$, $C_3^6.(C_2\times A_4)$

Order 17280: $C_2\times S_4\times A_6$ x 5, $A_4\times S_4\times A_5$ x 4, $A_5\times \PSOPlus(4,3)$ x 3, $A_6.C_2\times S_4$ x 2, $A_4\wr C_2\times A_5$ x 2, $C_2^2:A_4\times A_6$ x 2, $C_2^2\times A_4\times A_6$ x 2, $S_4\times \PGL(2,9)$, $S_4\times S_6$, $\GL(2,4):D_{12}:C_4$, $C_4\times A_4\times A_6$, $A_4:C_4\times A_6$, $S_3\times D_4\times A_6$

Order 16200: $C_3\times C_{15}\times A_6$

Order 15552: $C_3^4.A_4.D_4:C_2$ x 4, $C_3^4.(D_4\times S_4)$ x 2, $A_4.C_3^4.C_4^2$ x 2, $A_4.C_3^4.C_2.C_2^3$ x 2, $C_6^2.(S_3\times F_9)$, $C_3^4.Q_8:S_4$, $C_3^4.C_8:S_4$, $C_3^4.A_4.\OD_{16}$, $C_3^4.A_4.C_4^2$, $C_3^4.(Q_8\times S_4)$, $C_3^4.(A_4\times \OD_{16})$, $C_3^4.(A_4\times D_4:C_2)$, $A_4.C_3^4.C_2^3.C_2$, $C_3^4:C_4^2:D_6$

Order 14400: $C_2^2\times A_5^2$ x 3, $A_5^2:C_2^2$ x 3, $D_{10}.A_6.C_2$ x 2, $A_6.C_2\times F_5$ x 2, $D_5\times S_4\times A_5$ x 2, $C_4\times A_5^2$ x 2, $A_5^2:C_4$ x 2, $C_{15}^2.C_2^2.C_2^4$, $C_5:C_4.A_6.C_2$, $C_2\times D_5.A_6.C_2$, $\PGL(2,9):F_5$, $S_6:F_5$, $C_5\times D_4\times A_6$, $C_2\times D_{10}\times A_6$, $D_{20}\times A_6$, $C_4\times D_5\times A_6$, $C_5:D_4\times A_6$

Order 13824: $S_4^3$ x 2, $A_4^3.C_2^3$ x 2, $A_4^2:C_4\times S_4$ x 2, $A_4^2:\GL(2,\mathbb{Z}/4)$ x 2, $A_4^3:D_4$ x 2, $S_4^2:S_4$ x 2, $A_4^3:D_4$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 559872000: $A_6^3.D_6$ ($C_1$)

Order 279936000: $A_6^3.C_6$ ($C_2$), $A_6^3.S_3$ ($C_2$), $A_6\wr S_3$ ($C_2$)

Order 139968000: $A_6\wr C_3$ ($C_2^2$)

Order 93312000: $A_6.A_6.A_6.C_2$ ($S_3$)

Order 46656000: $A_6.A_6.A_6$ ($D_6$)

Order 1: $C_1$ ($A_6^3.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $112 \times 112$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $95 \times 95$ rational character table.