Abstract group 4199040.a: $C_3^4:\Sp(4,3)$

• Label: 4199040.a
• Order: \( 2^{7} \cdot 3^{8} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $81$
• Trans deg.: $81$
• Rank: $2$

Group information

Description:	$C_3^4:\Sp(4,3)$
Order:	\(4199040\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 5 \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \)
Composition factors:	$C_2$, $C_3$ x 4, $\SU(4,2)$
Derived length:	$0$

This group is nonabelian and perfect (hence nonsolvable).

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	18
Elements	1	891	64880	92340	419904	654480	524880	466560	419904	1088640	466560	4199040
Conjugacy classes	1	2	14	3	1	21	1	4	1	8	2	58
Divisions	1	2	9	3	1	12	1	2	1	5	1	38
Autjugacy classes	1	2	9	3	1	12	1	2	1	5	1	38

Dimension	1	4	5	6	8	10	15	20	24	30	36	40	45	60	64	72	80	81	90	120	160	240	320	480	640	720	960	1280	1440
Irr. complex chars.	1	2	2	1	0	2	2	6	1	3	2	2	2	4	2	0	4	1	0	0	3	7	0	6	3	2	0	0	0	58
Irr. rational chars.	1	0	0	1	1	1	2	3	1	1	0	2	0	3	2	1	3	1	1	1	2	1	1	3	1	0	3	1	1	38

Minimal presentations

Permutation degree:	$81$
Transitive degree:	$81$
Rank:	$2$
Inequivalent generating pairs:	$1840800$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	80	80	80
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\ASp(4,3)$, $\ASigmaSp(4,3)$
Permutation group:	Degree $81$ $\langle(2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,55)(29,57) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^4$ $\,\rtimes\,$ $\Sp(4,3)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:S_3)$ . $\SU(4,2)$				more information

Elements of the group are displayed as permutations of degree 81.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 11069772 subgroups in 1222 conjugacy classes, 4 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^4:\Sp(4,3)$
Commutator:	$G' \simeq$ $C_3^4:\Sp(4,3)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^4:\Sp(4,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4$	$G/\operatorname{Fit} \simeq$ $\Sp(4,3)$
Radical:	$R \simeq$ $C_3^3:S_3$	$G/R \simeq$ $\SU(4,2)$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $\Sp(4,3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^3.C_3$
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 4199040: $C_3^4:\Sp(4,3)$

Order 155520: $C_3^4.(C_4.C_2^3).A_5$

Order 116640: $\ASigmaL(2,9)$

Order 104976: $C_3^4.C_3^2.C_6.A_4.C_2$, $C_3^4.C_3^3.Q_8.S_3$

Order 93312: $\PU(3,2)\wr C_2$

Order 58320: $C_3^4.\SL(2,9)$

Order 52488: $C_3^4.C_3^3.Q_8.C_3$, $C_3^4.C_3:S_3.A_4.C_3$

Order 51840: $\Sp(4,3)$

Order 46656: $C_3^4.Q_8^2.C_3^2$

Order 34992: $C_3^4.C_3^2.D_6.C_2^2$, $C_3^4.C_3^2.Q_8.C_6$, $C_3^4.C_3^3.\SD_{16}$

Order 31104: $C_3^4:(Q_8^2:S_3)$, $C_3^4:(Q_8^2:C_6)$

Order 26244: $C_3^4.C_3^3.C_6.C_2$

Order 25920: $C_3^4.(C_4.C_2^3).D_5$

Order 19440: $C_3^4.(C_2.S_5)$, $C_3^4:\SL(2,5):C_2$

Order 17496: $C_3^2.C_3^4.Q_8.C_3$, $C_3^4.C_3^3.C_2^2.C_2$, $C_3^4.C_3^3.Q_8$, $C_3^5.F_9$, $C_3^4.C_3^3.Q_8$, $C_3^4:C_3.C_6^2.C_2$, $C_3^4.C_3:S_3.A_4$, $C_3^4.C_3:S_3.C_6.C_2$

Order 15552: $C_3^4.Q_8^2.C_3$ x 2, $C_3^4.Q_8.C_{12}.C_2$, $C_3^3:S_3.\GL(2,\mathbb{Z}/4)$

Order 13122: $C_3^4.C_3^3.C_6$, $C_3^4.C_3^3.C_6$, $C_3^4.C_3^3.C_6$

Order 12960: $C_3^4.(C_4.C_2^3).C_5$

Order 11664: $C_3^2.Q_8:\He_3.C_6$, $C_3^4.C_3:S_3.D_4$, $C_3^4.C_3^2.Q_8.C_2$, $C_3^5:\GL(2,3)$

Order 10368: $\PSU(3,2)\wr C_2$

Order 9720: $C_3^4:\SL(2,5)$ x 2

Order 8748: $C_3^4.C_3^3.C_4$, $C_3^3.\He_3.D_6$, $C_3^4.C_3^3.C_2^2$, $C_3^4:C_3.C_6^2$, $C_3^4.C_3^3.C_4$, $C_3^4.C_3^3.C_4$

Order 7776: $C_3^4:(C_4\times \SL(2,3))$, $C_3^4:C_4.S_4$, $C_3^4:\Unitary(2,3)$, $C_3^4:\GL(2,3):C_2$, $C_3^4:D_4.A_4$

Order 6561: $C_3^4.C_3^3.C_3$

Order 5832: $C_3:S_3.(A_4\times \He_3)$, $C_3^4.C_3^2.C_8$, $C_3^4.C_3^2.Q_8$, $C_3^4.C_3^2.Q_8$, $C_3^4.C_3^2.C_4.C_2$, $C_3^2.C_3^4.Q_8$, $C_3^4.D_6:S_3$, $C_3^5:\SL(2,3)$, $\He_3^2:D_4$

Order 5184: $C_3^4.Q_8^2$, $C_3^2.Q_8^2.C_3^2$, $C_3^4:D_4.D_4$, $C_3^4:C_4^2.C_4$

Order 4374: $C_3^4.C_3^3.C_2$, $C_3^3.\He_3:S_3$, $C_3^4.\He_3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^3.\He_3.C_6$, $C_3^4.\He_3.C_2$, $C_3^4.C_3^2.C_6$

Order 3888: $C_3:S_3.C_6^2.C_6$ x 2, $C_3^3:S_3.S_4$ x 2, $C_3^4:(C_2\times \SL(2,3))$ x 2, $C_3^4.(C_6\times Q_8)$, $\He_3.(C_3\times Q_8).C_6$, $S_3\times C_3^3:\SL(2,3)$, $C_3^3:(S_3\times \SL(2,3))$, $C_3^3:F_9:C_2$, $C_3^5:\SD_{16}$, $C_3^4:\GL(2,3)$, $C_3^4:\SL(2,3):C_2$

Order 3240: $C_3^4:C_5:C_8$

Order 2916: $C_3^4:C_3.C_6.C_2$, $C_3^4.C_6^2$, $C_3^4.C_3^2.C_4$, $C_3^2.C_3^4.C_4$, $C_3^2.C_3^4.C_4$, $C_3^4:C_3.C_{12}$, $C_3^4.C_3:S_3.C_2$, $C_3^4:C_3.C_6.C_2$, $C_3^5:D_6$, $\He_3^2:C_4$, $C_3^4:S_3^2$, $C_3^5:D_6$

Order 2592: $C_3^4:(C_4\times Q_8)$, $C_3^3:C_{12}:Q_8$, $C_3^4:C_4.D_4$, $C_3^4:Q_{16}:C_2$, $C_3^4:C_4.C_2^3$, $C_3^4:C_4\wr C_2$

Order 2187: $C_3^4.C_3^3$ x 2, $C_3^4.C_3^3$ x 2, $C_3^3.C_3^4$, $C_3^4.C_3^3$, $C_3^4.C_3^3$

Order 1944: $C_3^4:\SL(2,3)$ x 5, $C_3^4:\SL(2,3)$ x 3, $C_3^4:\SL(2,3)$ x 2, $C_3^5:Q_8$, $C_3^5:D_4$, $C_3^3:F_9$, $\He_3:\PSU(3,2)$, $\He_3\times \PSU(3,2)$, $C_3^4:(C_2\times C_{12})$, $C_3\times \Unitary(3,2)$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$

Order 1920: $C_2.C_2^4:A_5$

Order 1728: $\PSU(3,2):\SL(2,3)$ x 2, $\SL(2,3)\times \PSU(3,2)$, $Q_8\times \PU(3,2)$

Order 1620: $C_3^4:C_5:C_4$

Order 1458: $C_3^4:(C_3\times S_3)$ x 4, $(C_3^2\times \He_3):S_3$ x 3, $C_3^4:(C_3\times S_3)$ x 2, $\He_3^2:C_2$ x 2, $\He_3\wr C_2$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^5:S_3$ x 2, $C_3^4:C_{18}$, $C_3^4.(C_3\times S_3)$, $C_3^4:(C_3\times C_6)$, $\He_3^2:C_2$, $C_3^4:(C_3\times C_6)$, $(C_3^2\times \He_3):S_3$, $C_3^4:(C_3\times S_3)$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^4.(C_3\times C_6)$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1440: $\SL(2,9):C_2$

Order 1296: $S_3\times \PU(3,2)$ x 3, $C_3^4:(C_2\times Q_8)$ x 2, $C_3^4:D_4:C_2$, $C_3^3:(S_3\times Q_8)$, $S_3\times C_3^3:Q_8$, $C_3^4:Q_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\OD_{16}$, $C_3^2\wr C_2.Q_8$, $C_3^4:C_4^2$, $C_6\times \PU(3,2)$, $C_3^3:\GL(2,3)$, $S_3\times \SU(3,2)$, $C_2\times \Unitary(3,2)$, $C_3^2:C_6\times \SL(2,3)$

Order 1152: $\SL(2,3)\wr C_2$

Order 972: $C_3^4:D_6$ x 3, $C_3^3:S_3^2$ x 2, $C_3^4:C_3:C_4$ x 2, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_{12}$

Order 864: $C_3^2:C_4\times \SL(2,3)$, $C_4\times \PU(3,2)$

Order 810: $C_3^4:C_{10}$

Order 729: $C_3^4:C_3^2$ x 5, $C_3^4.C_3^2$ x 5, $\He_3^2$ x 3, $C_3^3.C_3^3$ x 3, $C_3^4:C_9$ x 2, $C_3^4:C_3^2$ x 2, $C_3^3.C_3^3$ x 2, $C_3^5:C_3$ x 2, $C_3^5:C_3$

Order 720: $\SL(2,9)$

Order 648: $C_3^3:\SL(2,3)$ x 11, $C_3^4:Q_8$ x 3, $C_3^4:Q_8$ x 2, $C_3:\SU(3,2)$ x 2, $\Unitary(3,2)$ x 2, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3\wr D_4$, $C_3^4:(C_2\times C_4)$, $C_3^3:(C_4\times S_3)$, $S_3\times C_3^2:C_{12}$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^3:\SL(2,3)$, $C_3\times \SU(3,2)$, $S_3\times \He_3:C_4$, $\He_3\times \SL(2,3)$

Order 576: $C_3^2:Q_8^2$, $\SL(2,3)^2$

Order 486: $C_3^4:C_6$ x 20, $C_3^4:S_3$ x 11, $C_3^4:C_6$ x 9, $(C_3\times \He_3):C_6$ x 3, $C_3^4:S_3$ x 3, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^3:C_{18}$, $C_3^4:C_6$, $C_3^4.C_6$

Order 432: $C_2\times \PU(3,2)$ x 5, $(C_3\times \SL(2,3)):S_3$ x 2, $C_6\times \PSU(3,2)$, $S_3\times \PSU(3,2)$, $C_3^3:\SD_{16}$, $C_3^2:\GL(2,3)$, $C_3:S_3\times \SL(2,3)$, $C_3\times S_3\times \SL(2,3)$, $C_3^3:\SD_{16}$, $C_2\times \SU(3,2)$, $C_3^2:C_6\times Q_8$

Order 405: $C_3^4:C_5$

Order 384: $Q_8^2:C_6$, $Q_8^2:S_3$

Order 324: $(C_3\times \He_3):C_4$ x 4, $C_3\wr C_2^2$ x 3, $C_3\wr C_4$ x 3, $C_3^3:C_{12}$ x 3, $C_3^2:S_3^2$ x 3, $C_3^2\times S_3^2$ x 2, $C_3^4:C_4$ x 2, $\He_3:C_{12}$ x 2, $C_3^3:D_6$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^4:C_4$, $C_3^2:C_6^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^3:D_6$

Order 320: $C_2.C_2^4:D_5$

Order 288: $C_3^2:C_4\times Q_8$, $C_4:\PSU(3,2)$, $C_4\times \PSU(3,2)$

Order 243: $C_3^4:C_3$ x 42, $C_3^2\times \He_3$ x 10, $C_3^3.C_3^2$ x 6, $C_3^3:C_9$ x 5, $C_3^4:C_3$ x 4, $C_3^4.C_3$ x 2, $C_3^5$

Order 240: $\SL(2,5):C_2$, $C_2.S_5$

Order 216: $\PU(3,2)$ x 9, $C_3^2:\SL(2,3)$ x 5, $C_3^3:Q_8$ x 4, $\SU(3,2)$ x 3, $C_3^2\times \SL(2,3)$ x 3, $C_3^3:Q_8$ x 2, $Q_8\times \He_3$, $C_4\times C_3^2:C_6$, $\He_3:Q_8$, $C_2\times C_3^2:C_{12}$, $S_3^2:S_3$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$, $C_3:F_9$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_3^3:C_8$, $C_2\times \He_3:C_4$

Order 192: $Q_8:\SL(2,3)$ x 2, $D_4.S_4$, $Q_8\times \SL(2,3)$

Order 162: $C_3^3:C_6$ x 44, $C_3^2\wr C_2$ x 16, $S_3\times \He_3$ x 8, $C_3\wr S_3$ x 7, $S_3\times C_3^3$ x 5, $C_3^3:S_3$ x 5, $C_3^3:C_6$ x 4, $C_3^3:S_3$ x 3, $C_3^3:S_3$, $C_3^3:C_6$, $C_6\times \He_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $C_3^3:C_6$

Order 160: $C_2.C_2^4:C_5$

Order 144: $C_2\times \PSU(3,2)$ x 3, $S_3\times \SL(2,3)$ x 3, $F_9:C_2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6\times \SL(2,3)$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $C_2.\PSU(3,2)$, $C_3^2:\SD_{16}$

Order 128: $Q_8\wr C_2$

Order 120: $\SL(2,5)$ x 2

Order 108: $C_3\times S_3^2$ x 7, $\He_3:C_4$ x 6, $C_3^2:C_{12}$ x 5, $C_3^3:C_4$ x 3, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 2, $C_3^2:C_{12}$, $C_3^2:D_6$, $C_3:S_3^2$, $C_3:S_3^2$, $C_3^2:C_{12}$, $C_3^2:D_6$, $C_3^2:D_6$, $C_4\times \He_3$

Order 96: $C_4\times \SL(2,3)$, $\Unitary(2,3)$, $D_4.A_4$, $C_4.S_4$, $\GL(2,3):C_2$

Order 81: $C_3\times \He_3$ x 69, $C_3^4$ x 14, $C_3\wr C_3$ x 10, $C_3^2:C_9$ x 5, $C_9:C_3^2$ x 2

Order 72: $C_3\times \SL(2,3)$ x 11, $\PSU(3,2)$ x 5, $Q_8\times C_3^2$ x 3, $C_2\times C_3^2:C_4$ x 2, $\SOPlus(4,2)$, $F_9$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_6\wr C_2$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $D_6:S_3$, $C_3^2:C_8$

Order 64: $C_4^2.C_4$, $Q_8^2$, $D_4.D_4$

Order 54: $S_3\times C_3^2$ x 32, $C_3^2:C_6$ x 26, $C_3^2:C_6$ x 18, $C_3^2:S_3$ x 7, $C_2\times \He_3$ x 7, $C_3^2\times C_6$ x 4, $S_3\times C_9$, $C_3^2:S_3$, $C_9:C_6$

Order 48: $C_2\times \SL(2,3)$ x 5, $C_2.S_4$ x 2, $C_6\times Q_8$, $S_3\times Q_8$, $\SL(2,3):C_2$, $\GL(2,3)$, $C_3:\SD_{16}$

Order 40: $C_5:C_8$

Order 36: $C_6\times S_3$ x 7, $C_3^2:C_4$ x 4, $C_3\times C_{12}$ x 3, $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $S_3^2$ x 2, $C_6^2$, $C_6:S_3$

Order 32: $C_4.D_4$, $C_4.C_2^3$, $Q_{16}:C_2$, $C_4:Q_8$, $C_4\times Q_8$, $C_4\wr C_2$

Order 27: $\He_3$ x 50, $C_3^3$ x 42, $C_9:C_3$ x 5, $C_3\times C_9$ x 2

Order 24: $\SL(2,3)$ x 8, $C_3\times Q_8$ x 4, $C_3:Q_8$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $C_4\times S_3$, $C_3\times D_4$, $C_3:C_8$

Order 20: $C_5:C_4$

Order 18: $C_3\times S_3$ x 28, $C_3\times C_6$ x 17, $C_3:S_3$ x 5, $C_{18}$

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

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 3, $C_3:C_4$ x 3, $D_6$ x 2

Order 10: $C_{10}$

Order 9: $C_3^2$ x 44, $C_9$ x 2

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

Order 6: $C_6$ x 12, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_4$ x 3, $C_2^2$

Order 3: $C_3$ x 9

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4199040: $C_3^4:\Sp(4,3)$

Order 155520: $C_3^4.(C_4.C_2^3).A_5$

Order 116640: $\ASigmaL(2,9)$

Order 104976: $C_3^4.C_3^2.C_6.A_4.C_2$, $C_3^4.C_3^3.Q_8.S_3$

Order 93312: $\PU(3,2)\wr C_2$

Order 58320: $C_3^4.\SL(2,9)$

Order 52488: $C_3^4.C_3^3.Q_8.C_3$, $C_3^4.C_3:S_3.A_4.C_3$

Order 51840: $\Sp(4,3)$

Order 46656: $C_3^4.Q_8^2.C_3^2$

Order 34992: $C_3^4.C_3^2.D_6.C_2^2$, $C_3^4.C_3^2.Q_8.C_6$, $C_3^4.C_3^3.\SD_{16}$

Order 31104: $C_3^4:(Q_8^2:S_3)$, $C_3^4:(Q_8^2:C_6)$

Order 26244: $C_3^4.C_3^3.C_6.C_2$

Order 25920: $C_3^4.(C_4.C_2^3).D_5$

Order 19440: $C_3^4.(C_2.S_5)$, $C_3^4:\SL(2,5):C_2$

Order 17496: $C_3^2.C_3^4.Q_8.C_3$, $C_3^4.C_3^3.C_2^2.C_2$, $C_3^4.C_3^3.Q_8$, $C_3^5.F_9$, $C_3^4.C_3^3.Q_8$, $C_3^4:C_3.C_6^2.C_2$, $C_3^4.C_3:S_3.A_4$, $C_3^4.C_3:S_3.C_6.C_2$

Order 15552: $C_3^4.Q_8^2.C_3$ x 2, $C_3^4.Q_8.C_{12}.C_2$, $C_3^3:S_3.\GL(2,\mathbb{Z}/4)$

Order 13122: $C_3^4.C_3^3.C_6$, $C_3^4.C_3^3.C_6$, $C_3^4.C_3^3.C_6$

Order 12960: $C_3^4.(C_4.C_2^3).C_5$

Order 11664: $C_3^2.Q_8:\He_3.C_6$, $C_3^4.C_3:S_3.D_4$, $C_3^4.C_3^2.Q_8.C_2$, $C_3^5:\GL(2,3)$

Order 10368: $\PSU(3,2)\wr C_2$

Order 9720: $C_3^4:\SL(2,5)$ x 2

Order 8748: $C_3^4.C_3^3.C_4$, $C_3^3.\He_3.D_6$, $C_3^4.C_3^3.C_2^2$, $C_3^4:C_3.C_6^2$, $C_3^4.C_3^3.C_4$, $C_3^4.C_3^3.C_4$

Order 7776: $C_3^4:(C_4\times \SL(2,3))$, $C_3^4:C_4.S_4$, $C_3^4:\Unitary(2,3)$, $C_3^4:\GL(2,3):C_2$, $C_3^4:D_4.A_4$

Order 6561: $C_3^4.C_3^3.C_3$

Order 5832: $C_3:S_3.(A_4\times \He_3)$, $C_3^4.C_3^2.C_8$, $C_3^4.C_3^2.Q_8$, $C_3^4.C_3^2.Q_8$, $C_3^4.C_3^2.C_4.C_2$, $C_3^2.C_3^4.Q_8$, $C_3^4.D_6:S_3$, $C_3^5:\SL(2,3)$, $\He_3^2:D_4$

Order 5184: $C_3^4.Q_8^2$, $C_3^2.Q_8^2.C_3^2$, $C_3^4:D_4.D_4$, $C_3^4:C_4^2.C_4$

Order 4374: $C_3^4.C_3^3.C_2$, $C_3^3.\He_3:S_3$, $C_3^4.\He_3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^3.\He_3.C_6$, $C_3^4.\He_3.C_2$, $C_3^4.C_3^2.C_6$

Order 3888: $C_3:S_3.C_6^2.C_6$ x 2, $C_3^3:S_3.S_4$ x 2, $C_3^4:(C_2\times \SL(2,3))$ x 2, $C_3^4.(C_6\times Q_8)$, $\He_3.(C_3\times Q_8).C_6$, $S_3\times C_3^3:\SL(2,3)$, $C_3^3:(S_3\times \SL(2,3))$, $C_3^3:F_9:C_2$, $C_3^5:\SD_{16}$, $C_3^4:\GL(2,3)$, $C_3^4:\SL(2,3):C_2$

Order 3240: $C_3^4:C_5:C_8$

Order 2916: $C_3^4:C_3.C_6.C_2$, $C_3^4.C_6^2$, $C_3^4.C_3^2.C_4$, $C_3^2.C_3^4.C_4$, $C_3^2.C_3^4.C_4$, $C_3^4:C_3.C_{12}$, $C_3^4.C_3:S_3.C_2$, $C_3^4:C_3.C_6.C_2$, $C_3^5:D_6$, $\He_3^2:C_4$, $C_3^4:S_3^2$, $C_3^5:D_6$

Order 2592: $C_3^4:(C_4\times Q_8)$, $C_3^3:C_{12}:Q_8$, $C_3^4:C_4.D_4$, $C_3^4:Q_{16}:C_2$, $C_3^4:C_4.C_2^3$, $C_3^4:C_4\wr C_2$

Order 2187: $C_3^4.C_3^3$ x 2, $C_3^4.C_3^3$ x 2, $C_3^3.C_3^4$, $C_3^4.C_3^3$, $C_3^4.C_3^3$

Order 1944: $C_3^4:\SL(2,3)$ x 5, $C_3^4:\SL(2,3)$ x 3, $C_3^4:\SL(2,3)$ x 2, $C_3^5:Q_8$, $C_3^5:D_4$, $C_3^3:F_9$, $\He_3:\PSU(3,2)$, $\He_3\times \PSU(3,2)$, $C_3^4:(C_2\times C_{12})$, $C_3\times \Unitary(3,2)$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$

Order 1920: $C_2.C_2^4:A_5$

Order 1728: $\PSU(3,2):\SL(2,3)$ x 2, $\SL(2,3)\times \PSU(3,2)$, $Q_8\times \PU(3,2)$

Order 1620: $C_3^4:C_5:C_4$

Order 1458: $C_3^4:(C_3\times S_3)$ x 4, $(C_3^2\times \He_3):S_3$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $\He_3^2:C_2$ x 2, $\He_3\wr C_2$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^5:S_3$ x 2, $C_3^4:C_{18}$, $C_3^4.(C_3\times S_3)$, $C_3^4:(C_3\times C_6)$, $\He_3^2:C_2$, $C_3^4:(C_3\times C_6)$, $(C_3^2\times \He_3):S_3$, $C_3^4:(C_3\times S_3)$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^4.(C_3\times C_6)$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1440: $\SL(2,9):C_2$

Order 1296: $S_3\times \PU(3,2)$ x 3, $C_3^4:(C_2\times Q_8)$ x 2, $C_3^4:D_4:C_2$, $C_3^3:(S_3\times Q_8)$, $S_3\times C_3^3:Q_8$, $C_3^4:Q_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\OD_{16}$, $C_3^2\wr C_2.Q_8$, $C_3^4:C_4^2$, $C_6\times \PU(3,2)$, $C_3^3:\GL(2,3)$, $S_3\times \SU(3,2)$, $C_2\times \Unitary(3,2)$, $C_3^2:C_6\times \SL(2,3)$

Order 1152: $\SL(2,3)\wr C_2$

Order 972: $C_3^4:D_6$ x 3, $C_3^3:S_3^2$ x 2, $C_3^4:C_3:C_4$ x 2, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_{12}$

Order 864: $C_3^2:C_4\times \SL(2,3)$, $C_4\times \PU(3,2)$

Order 810: $C_3^4:C_{10}$

Order 729: $C_3^4:C_3^2$ x 5, $C_3^4.C_3^2$ x 4, $\He_3^2$ x 3, $C_3^4:C_9$ x 2, $C_3^4:C_3^2$ x 2, $C_3^3.C_3^3$ x 2, $C_3^5:C_3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^5:C_3$

Order 720: $\SL(2,9)$

Order 648: $C_3^3:\SL(2,3)$ x 10, $C_3^4:Q_8$ x 3, $C_3^4:Q_8$ x 2, $C_3:\SU(3,2)$ x 2, $\Unitary(3,2)$ x 2, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3\wr D_4$, $C_3^4:(C_2\times C_4)$, $C_3^3:(C_4\times S_3)$, $S_3\times C_3^2:C_{12}$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^3:\SL(2,3)$, $C_3\times \SU(3,2)$, $S_3\times \He_3:C_4$, $\He_3\times \SL(2,3)$

Order 576: $C_3^2:Q_8^2$, $\SL(2,3)^2$

Order 486: $C_3^4:C_6$ x 20, $C_3^4:S_3$ x 11, $C_3^4:C_6$ x 9, $C_3^4:S_3$ x 3, $(C_3\times \He_3):C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^3:C_{18}$, $C_3^4:C_6$, $C_3^4.C_6$

Order 432: $C_2\times \PU(3,2)$ x 5, $(C_3\times \SL(2,3)):S_3$ x 2, $C_6\times \PSU(3,2)$, $S_3\times \PSU(3,2)$, $C_3^3:\SD_{16}$, $C_3^2:\GL(2,3)$, $C_3:S_3\times \SL(2,3)$, $C_3\times S_3\times \SL(2,3)$, $C_3^3:\SD_{16}$, $C_2\times \SU(3,2)$, $C_3^2:C_6\times Q_8$

Order 405: $C_3^4:C_5$

Order 384: $Q_8^2:C_6$, $Q_8^2:S_3$

Order 324: $C_3\wr C_2^2$ x 3, $C_3\wr C_4$ x 3, $C_3^3:C_{12}$ x 3, $C_3^2:S_3^2$ x 3, $(C_3\times \He_3):C_4$ x 3, $C_3^2\times S_3^2$ x 2, $C_3^4:C_4$ x 2, $\He_3:C_{12}$ x 2, $C_3^3:D_6$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^4:C_4$, $C_3^2:C_6^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^3:D_6$

Order 320: $C_2.C_2^4:D_5$

Order 288: $C_3^2:C_4\times Q_8$, $C_4:\PSU(3,2)$, $C_4\times \PSU(3,2)$

Order 243: $C_3^4:C_3$ x 42, $C_3^2\times \He_3$ x 10, $C_3^3:C_9$ x 5, $C_3^4:C_3$ x 4, $C_3^3.C_3^2$ x 4, $C_3^4.C_3$ x 2, $C_3^5$

Order 240: $\SL(2,5):C_2$, $C_2.S_5$

Order 216: $\PU(3,2)$ x 9, $C_3^2:\SL(2,3)$ x 5, $C_3^3:Q_8$ x 4, $\SU(3,2)$ x 3, $C_3^2\times \SL(2,3)$ x 3, $C_3^3:Q_8$ x 2, $Q_8\times \He_3$, $C_4\times C_3^2:C_6$, $\He_3:Q_8$, $C_2\times C_3^2:C_{12}$, $S_3^2:S_3$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$, $C_3:F_9$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_3^3:C_8$, $C_2\times \He_3:C_4$

Order 192: $Q_8:\SL(2,3)$ x 2, $D_4.S_4$, $Q_8\times \SL(2,3)$

Order 162: $C_3^3:C_6$ x 38, $C_3^2\wr C_2$ x 16, $S_3\times \He_3$ x 8, $C_3\wr S_3$ x 6, $S_3\times C_3^3$ x 5, $C_3^3:S_3$ x 5, $C_3^3:C_6$ x 4, $C_3^3:S_3$ x 3, $C_3^3:S_3$, $C_3^3:C_6$, $C_6\times \He_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $C_3^3:C_6$

Order 160: $C_2.C_2^4:C_5$

Order 144: $C_2\times \PSU(3,2)$ x 3, $S_3\times \SL(2,3)$ x 3, $F_9:C_2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6\times \SL(2,3)$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $C_2.\PSU(3,2)$, $C_3^2:\SD_{16}$

Order 128: $Q_8\wr C_2$

Order 120: $\SL(2,5)$ x 2

Order 108: $C_3\times S_3^2$ x 7, $C_3^2:C_{12}$ x 5, $\He_3:C_4$ x 5, $C_3^3:C_4$ x 3, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 2, $C_3^2:C_{12}$, $C_3^2:D_6$, $C_3:S_3^2$, $C_3:S_3^2$, $C_3^2:C_{12}$, $C_3^2:D_6$, $C_3^2:D_6$, $C_4\times \He_3$

Order 96: $C_4\times \SL(2,3)$, $\Unitary(2,3)$, $D_4.A_4$, $C_4.S_4$, $\GL(2,3):C_2$

Order 81: $C_3\times \He_3$ x 63, $C_3^4$ x 14, $C_3\wr C_3$ x 8, $C_3^2:C_9$ x 4, $C_9:C_3^2$ x 2

Order 72: $C_3\times \SL(2,3)$ x 10, $\PSU(3,2)$ x 5, $Q_8\times C_3^2$ x 3, $C_2\times C_3^2:C_4$ x 2, $\SOPlus(4,2)$, $F_9$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_6\wr C_2$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $D_6:S_3$, $C_3^2:C_8$

Order 64: $C_4^2.C_4$, $Q_8^2$, $D_4.D_4$

Order 54: $S_3\times C_3^2$ x 31, $C_3^2:C_6$ x 25, $C_3^2:C_6$ x 18, $C_3^2:S_3$ x 7, $C_2\times \He_3$ x 7, $C_3^2\times C_6$ x 4, $S_3\times C_9$, $C_3^2:S_3$, $C_9:C_6$

Order 48: $C_2\times \SL(2,3)$ x 5, $C_2.S_4$ x 2, $C_6\times Q_8$, $S_3\times Q_8$, $\SL(2,3):C_2$, $\GL(2,3)$, $C_3:\SD_{16}$

Order 40: $C_5:C_8$

Order 36: $C_6\times S_3$ x 7, $C_3^2:C_4$ x 4, $C_3\times C_{12}$ x 3, $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $S_3^2$ x 2, $C_6^2$, $C_6:S_3$

Order 32: $C_4.D_4$, $C_4.C_2^3$, $Q_{16}:C_2$, $C_4:Q_8$, $C_4\times Q_8$, $C_4\wr C_2$

Order 27: $\He_3$ x 48, $C_3^3$ x 41, $C_9:C_3$ x 5, $C_3\times C_9$ x 2

Order 24: $\SL(2,3)$ x 8, $C_3\times Q_8$ x 4, $C_3:Q_8$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $C_4\times S_3$, $C_3\times D_4$, $C_3:C_8$

Order 20: $C_5:C_4$

Order 18: $C_3\times S_3$ x 28, $C_3\times C_6$ x 16, $C_3:S_3$ x 5, $C_{18}$

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

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 3, $C_3:C_4$ x 3, $D_6$ x 2

Order 10: $C_{10}$

Order 9: $C_3^2$ x 42, $C_9$ x 2

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

Order 6: $C_6$ x 12, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_4$ x 3, $C_2^2$

Order 3: $C_3$ x 9

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4199040: $C_3^4:\Sp(4,3)$ ($C_1$)

Order 162: $C_3^3:S_3$ ($\SU(4,2)$)

Order 81: $C_3^4$ ($\Sp(4,3)$)

Order 1: $C_1$ ($C_3^4:\Sp(4,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4199040: $C_3^4:\Sp(4,3)$ ($C_1$)

Order 162: $C_3^3:S_3$ ($\SU(4,2)$)

Order 81: $C_3^4$ ($\Sp(4,3)$)

Order 1: $C_1$ ($C_3^4:\Sp(4,3)$)

Series

Derived series

$C_3^4:\Sp(4,3)$

$\rhd$

$C_3^4:\Sp(4,3)$

Chief series

$C_3^4:\Sp(4,3)$

$\rhd$

$C_3^4:\Sp(4,3)$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^4:\Sp(4,3)$

$\rhd$

$C_3^4:\Sp(4,3)$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 1 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 $58 \times 58$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.