Abstract group 47377612800.a: $\Sp(8,2)$

• Label: 47377612800.a
• Order: \( 2^{16} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 17 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $120$
• Trans deg.: $120$
• Rank: $2$

Group information

Description:	$\Sp(8,2)$
Order:	\(47377612800\)\(\medspace = 2^{16} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 17 \)
Exponent:	\(42840\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17 \)
Automorphism group:	$\Sp(8,2)$, of order \(47377612800\)\(\medspace = 2^{16} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 17 \)
Composition factors:	$\Sp(8,2)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	14	15	17	18	20	21	24	30
Elements	1	1371135	16461440	345461760	171085824	1830353280	1128038400	3948134400	2632089600	3355914240	8389785600	3384115200	4211343360	5573836800	2632089600	2368880640	2256076800	1974067200	3158507520	47377612800
Conjugacy classes	1	6	4	12	2	16	1	6	2	4	13	1	3	2	1	2	1	2	2	81
Divisions	1	6	4	12	2	16	1	6	2	4	13	1	3	1	1	2	1	2	2	80
Autjugacy classes	1	6	4	12	2	16	1	6	2	4	13	1	3	2	1	2	1	2	2	81

Dimension	1	35	51	85	119	135	238	510	595	918	1190	1275	1512	1785	2295	2856	2975	3213	3400	3570	3808	4200	4760	5712	5950	7140	8160	8925	8960	9639	10200	11900	13056	13600	14280	14688	16065	17850	18360	19040	23800	26775	28560	28917	30464	32130	34425	34560	38080	38556	42525	43520	47600	48195	51408	53550	57120	65536	68850	85050
Irr. complex chars.	1	1	1	1	1	1	1	1	2	1	1	1	1	2	1	2	2	2	1	3	1	1	1	1	2	1	1	2	1	1	1	2	1	1	2	1	5	2	1	1	2	2	1	1	1	3	2	1	1	1	2	1	1	1	1	1	1	1	1	0	81
Irr. rational chars.	1	1	1	1	1	1	1	1	2	1	1	1	1	2	1	2	2	2	1	3	1	1	1	1	2	1	1	2	1	1	1	2	1	1	2	1	5	2	1	1	2	2	1	1	1	3	2	1	1	1	0	1	1	1	1	1	1	1	1	1	80

Minimal presentations

Permutation degree:	$120$
Transitive degree:	$120$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	35	35	35
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\Sp(8,2)$, $\SO(9,2)$, $\PSp(8,2)$, $\Orth(9,2)$, $\Omega(9,2)$, $\PO(9,2)$, $\GSp(8,2)$, $\PSigmaSp(8,2)$
Permutation group:	Degree $120$ $\langle(1,3,6,11,17,23,30,39)(2,4,8,14,19,24,32,42)(5,9,15,20,26,34,44,36)(7,12,18,13,10,16,21,28) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\Sp(8,2)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 2 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 $\Sp(8,2)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.C_2^6.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

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 1880064 (order at least 25200) are shown, as well as all normal subgroups of any index.

Order 47377612800: $\Sp(8,2)$

Order 394813440: $\SOMinus(8,2)$

Order 348364800: $\SOPlus(8,2)$

Order 197406720: $\OmegaMinus(8,2)$

Order 185794560: $C_2^7.C(3,2)$

Order 174182400: $\OmegaPlus(8,2)$

Order 20643840: $C_2^{10}.A_8$

Order 8847360: $C_2^3.C_2^8.S_3.A_6.C_2$

Order 8709120: $S_3.C(3,2)$

Order 6635520: $C_2^7.\SU(4,2).C_2$

Order 5160960: $C_2^7.A_8.C_2$

Order 4423680: $C_2^3.C_2^8.C_3.A_6.C_2$ x 2, $C_2^3.C_2^8.S_3.A_6$

Order 4354560: $C_3\times C(3,2)$

Order 4128768: $C_2^3.C_2^6.S_3.C_2^3.\PSL(2,7)$

Order 3628800: $S_{10}$

Order 3317760: $C_2^6.\SU(4,2).C_2$ x 2, $C_2^7.\SU(4,2)$

Order 2949120: $C_2^7.C_2^5.A_6.C_2$

Order 2903040: $C_2\times C(3,2)$ x 2

Order 2580480: $C_2^6.A_8.C_2$ x 2, $C_2^{10}.A_7$, $C_2^7.A_8$

Order 2211840: $C_2^3.C_2^8.C_3.A_6$

Order 2064384: $C_2^3.C_2^6.C_3.C_2^3.\PSL(2,7)$

Order 1958400: $\PSigmaSp(4,4)$

Order 1814400: $A_{10}$

Order 1658880: $C_2^6.\SU(4,2)$

Order 1548288: $C_2^7.\SU(3,3).C_2$

Order 1474560: $C_2^7.C_2^4.A_6.C_2$ x 2, $C_2^7.C_2^5.A_6$, $C_2^3.C_2^8.S_3.S_5$, $C_2.C_2^8.S_4.S_5$

Order 1451520: $\Sp(6,2)$ x 3

Order 1376256: $C_2^7.C_2^6.\PSL(2,7)$ x 2

Order 1290240: $C_2^6.A_8$ x 2

Order 1036800: $S_6\wr C_2$

Order 979200: $\Sp(4,4)$

Order 884736: $C_2.C_2^8.C_6^2.C_6.C_2^3$

Order 774144: $C_2^7.\SU(3,3)$

Order 737280: $C_2.C_2^8.A_4.S_5$ x 2, $C_2^7.C_2^4.C_3.S_5$, $C_2^7.C_2^4.A_6$, $C_2^3.C_2^8.S_3.A_5$, $C_2^3.C_2^8.C_3.S_5$, $C_2.C_2^8.S_4.A_5$, $C_2^{10}.S_6$

Order 688128: $C_2^9.C_2^3.\PSL(2,7)$

Order 645120: $C_2^7.S_7$

Order 589824: $C_2^9.A_4^2.C_2^3$, $C_2^8.C_2^6.S_3^2$, $C_2.C_2^6.C_2^6.C_3^2.C_2^3$

Order 552960: $S_4\times C_2^5:S_6$

Order 518400: $S_6^2$, $A_6^2:C_4$, $A_6^2:C_2^2$

Order 516096: $C_2^3.C_2^6.S_3.\PSL(2,7)$ x 2, $C_2^3.C_2^6.F_8.C_3.C_6$

Order 491520: $C_2^7.C_2^5.S_5$ x 2

Order 442368: $Q_8:C_2^2.C_2^6.S_3^3$, $C_2^3.C_2^8.C_3^3.C_2^2.C_2$, $C_2.C_2^8.C_6^2.D_{12}$, $C_2.C_2^8.C_6^2.D_6.C_2$, $C_2.C_2^8.C_6^2.C_3.D_4$, $C_2.C_2^8.C_2^2.S_3^3$, $C_2.C_2^8.(A_4\times S_3^2).C_2$

Order 368640: $C_2^7.C_2^4.C_3.A_5$, $C_2^6:C_3.C_2^4.S_5$, $C_2^3.C_2^8.C_3.A_5$, $C_2.C_2^8.S_3.S_5$, $C_2^{10}.A_6$

Order 362880: $S_9$

Order 322560: $C_2^6.S_7$ x 2, $C_2^7.A_7$

Order 311040: $S_3.\SU(4,2).C_2$

Order 294912: $C_2.C_2^6.C_2^6.S_3^2$ x 4, $C_2^9.A_4\wr C_2.C_2$ x 3, $C_2^7.C_2^6.S_3^2$ x 2, $C_2^9.A_4^2.C_2^2$, $C_2^8.C_2^6.C_3.S_3$, $C_2^7.C_2^4.(C_6\times A_4).C_2$, $C_2^6.C_2^6.A_4.C_6$, $C_2^5.C_2^5.C_3.C_2^4.C_6$, $C_2^5.C_2^4.A_4^2.C_2^2$, $C_2^5.A_4^2.D_4^2$, $C_2.C_2^6.C_2^6.C_3.D_6$, $C_2.C_2^6.A_4.C_2^4.C_6.C_2$

Order 276480: $(A_4\times C_2^5).S_6$, $S_4\times C_2^5:A_6$, $(A_4\times C_2^5).S_6$

Order 259200: $A_6:S_6$, $A_6\times S_6$, $A_6\wr C_2$

Order 258048: $C_2^3.C_2^6.C_3.\PSL(2,7)$ x 2, $C_2^3.C_2^6.F_8.C_3^2$

Order 245760: $C_2^7.C_2^4.S_5$ x 4, $C_2^6.C_2^5.S_5$ x 4, $C_2^7.C_2^5.A_5$, $C_2^6.C_2^6.A_5$, $C_2.C_2^8.(F_5\times S_4)$

Order 241920: $S_3\times S_8$

Order 221184: $Q_8:C_2^2.C_2^6.C_3^2.D_6$ x 3, $Q_8:C_2^2.C_2^4.(A_4\times S_3^2)$, $Q_8.C_2^6.C_6^2.D_6$, $C_2^3.C_2^8.C_3^3.C_4$, $C_2^3.C_2^8.C_3^3.C_2^2$, $C_2^3.C_2^8.C_3^2.D_6$, $C_2.C_2^8.S_3^3.C_2$, $C_2.C_2^8.C_6^2.C_{12}$, $C_2.C_2^8.(A_4\times S_3^2)$, $C_2.C_2^6.C_2^4.C_3^3.C_2^2$

Order 196608: $C_2^9.C_2^4.D_6.C_2$, $C_2^7.C_2^4.C_6.C_2^4$, $C_2^6.C_2^6.A_4.C_2^2$, $C_2.C_2^6.A_4.C_2^4.C_2^3$

Order 193536: $C_2^7.\SL(2,8).C_3$

Order 184320: $C_2^7.C_2.A_6.C_2$ x 2, $C_2.C_2^8.C_3.S_5$ x 2, $C_2^6:C_3.C_2^4.A_5$, $C_2.C_2^8.S_3.A_5$

Order 181440: $A_9$ x 2

Order 172032: $C_2^{10}.\PSL(2,7)$ x 2, $C_2^7.C_2^3.\PSL(2,7)$ x 2, $C_2^6.C_2^4.\PSL(2,7)$ x 2, $C_2^6.C_2^4:\GL(3,2)$ x 2, $C_2^9.F_8.C_6$, $C_2^7.C_2^6.C_7.C_3$, $C_2^3.C_2^6.(S_3\times F_8)$, $C_2^{10}.\PSL(2,7)$

Order 165888: $S_4^3.D_6$, $C_2^7:(\He_3:\GL(2,3))$

Order 161280: $C_2^6:A_7$ x 2

Order 155520: $C_3.\SU(4,2).C_2$ x 3, $S_3\times \SU(4,2)$

Order 147456: $C_2^5.C_2^6.S_3^2.C_2$ x 4, $C_2^5.C_2^4.D_6\wr C_2$ x 4, $C_2^5.A_4^2.C_2^2:D_4$ x 4, $C_2^9.A_4^2.C_2$ x 2, $C_2^9.A_4\wr C_2$ x 2, $C_2^7.C_2^4.(S_3\times A_4)$ x 2, $C_2^5.A_4^2.(C_4\times D_4)$ x 2, $C_2^3.C_2^6.A_4^2.C_2$ x 2, $C_2.C_2^6.C_2^6.C_3.S_3$ x 2, $C_2.C_2^6.A_4.C_2^4.C_6$ x 2, $C_2^8.C_2^6.C_3^2$, $C_2^7.C_2^6.C_3.S_3$, $C_2^7.A_4^2.C_2^3$, $C_2^5.C_2^4.A_4^2.C_2$, $C_2^5.A_4^2.D_4.C_2^2$, $C_2^5.A_4^2.C_4:D_4$, $C_2^5.A_4^2.C_2^4.C_2$, $C_2^5.A_4^2.C_2.C_2^4$, $C_2^3.C_2^6.D_6^2.C_2$

Order 138240: $S_3\times C_2^5:S_6$, $A_4\times C_2^5:A_6$

Order 129600: $A_6^2$

Order 122880: $C_2^6.C_2^4.S_5$ x 4, $C_2^7.C_2^4.A_5$ x 2, $C_2^6.C_2^5.A_5$ x 2, $C_2^5.C_2^5.S_5$ x 2, $C_2^5.C_2^6.C_3:F_5$, $C_2^3.C_2^8.C_5.D_6$, $C_2.C_2^8.(A_4\times F_5)$, $C_2^{10}.S_5$, $C_2^{10}.S_5$

Order 120960: $S_3\times A_8$, $C_3:S_8$, $C_3\times S_8$

Order 110592: $C_2.C_2^8.C_3^3.D_4$ x 3, $Q_8:C_2^2.C_2^4.C_6^2.C_6$ x 2, $C_2.C_2^6.A_4^2.C_6$ x 2, $Q_8:C_2^2.C_2^4.S_3^3$, $C_2^3:A_4.C_2^6.C_3.C_6$, $C_2^3.C_2^6.C_3^2:S_4$, $C_2.C_2^8.C_3^3.C_2^2.C_2$, $C_2.C_2^8.C_3:S_3.C_6.C_2$, $C_2.C_2^6.C_2^4.C_3^3.C_2$, $D_4.C_2^6.S_3^3$

Order 103680: $C_2\times \SO(5,3)$

Order 98304: $C_2^7.C_2^4.C_6.C_2^3$ x 10, $C_2.C_2^6.A_4.C_2\wr C_2^2$ x 7, $C_2^9.C_2^4.D_6$ x 5, $C_2^5.C_2^4.C_6.C_2^4.C_2$ x 5, $C_2^5.C_2^6.C_6.C_2^3$ x 4, $C_2^9.C_2^4.C_6.C_2$, $C_2^9.A_4.C_2^3.C_2$, $C_2^8.C_2^6.S_3$, $C_2^8.C_2^4.C_6.C_2^2$, $C_2^8.C_2^4.C_3.D_4$, $C_2^7.C_2^6.D_6$, $C_2^6.C_2^6.S_4$, $C_2^6.C_2^6.A_4.C_2$, $C_2^5.C_2^5.A_4.C_2^3$, $C_2^5.C_2^4.D_6.C_2^4$, $C_2.C_2^6.C_2^6.D_6$, $C_2.C_2^6.A_4.C_2^5.C_2$

Order 92160: $(D_4\times C_2^4).S_6$ x 8, $C_2^7.A_6.C_2$ x 7, $(C_2^5\times C_4).A_6.C_2$ x 4, $C_2^7.C_2.A_6$ x 2, $C_2.C_2^8.C_3.A_5$, $S_4\times C_2^5:S_5$, $C_2\wr S_5.S_4$, $C_2^7.S_6$

Order 86400: $S_5\times S_6$ x 2

Order 86016: $C_2^6.C_2^3.\PSL(2,7)$ x 5, $C_2^9.\GL(3,2)$ x 2, $C_2^9.F_8.C_3$, $C_2^6.C_2^6.C_{21}$, $C_2^6.C_2^6.C_7.C_3$

Order 82944: $A_4^3.(C_2\times S_4)$ x 4, $C_2^6:\He_3:\GL(2,3)$ x 2, $C_2^6.\He_3.Q_8.C_6$, $C_2\times A_4^3:S_4$, $C_2\times A_4^3:S_4$, $S_4^3.C_6$

Order 81920: $C_2^5.C_2^4.(D_4\times F_5)$

Order 80640: $C_2\times S_8$ x 2

Order 77760: $\Unitary(4,2)$ x 2

Order 73728: $C_2^5.A_4^2.C_2^3.C_2$ x 22, $C_2^5.C_2^6.S_3^2$ x 11, $C_2^5.A_4^2.C_4.C_2^2$ x 5, $C_2.C_2^6.A_4^2.C_2^2$ x 5, $C_2^3.C_2^6.A_4.D_6$ x 4, $C_2^7.A_4\wr C_2.C_2$ x 3, $C_2^5.C_2^6.C_3.D_6$ x 3, $C_2^5.C_2^4.D_6:D_6$ x 3, $C_2^5.C_2^4.(S_3\times A_4).C_2$ x 3, $C_2^5.A_4^2.D_4.C_2$ x 3, $C_2^5.A_4^2.C_2^2:C_4$ x 3, $C_2^7.A_4^2.C_2^2$ x 2, $C_2^5.C_2^6.(C_6\times S_3)$ x 2, $C_2^5.A_4^2.C_4:C_4$ x 2, $C_2^5.A_4^2.C_2^4$ x 2, $C_2.C_2^6.C_2^4.S_3^2$ x 2, $C_2^9.A_4^2$, $C_2^9.A_4.D_6$, $C_2^6.C_2^6.C_3.S_3$, $C_2^5.C_2^5.(S_3\times A_4)$, $C_2^5.C_2^4.(C_6\times A_4).C_2$, $C_2^5.A_4^2.C_4^2$, $C_2^5.(C_2\times A_4^2).C_2^3$, $C_2^3:A_4.C_2^6.C_6.C_2$, $C_2^3:A_4.C_2^5.C_6.C_2^2$, $C_2^3.C_2^6.D_6^2$, $C_2^3.C_2^6.D_6:D_6$, $C_2^3.C_2^6.C_3.C_6.C_2^3$, $C_2^3.C_2^6.A_4^2$, $C_2^3.C_2^6.(S_3\times D_{12})$, $C_2.C_2^6.A_4\wr C_2.C_2$, $C_2^7.\POPlus(4,3)$, $C_2^{10}.\SOPlus(4,2)$

Order 72576: $S_3.\SU(3,3).C_2$

Order 69120: $(C_2^4\times C_6):S_6$, $S_3\times C_2^5:A_6$, $C_3\times C_2^5:S_6$

Order 65536: $C_2^8.C_2^6.C_2^2$

Order 64512: $C_2^7.\SL(2,8)$, $C_2^3.C_2^6.C_{21}.C_6$, $S_3\times C_2^6:\GL(3,2)$

Order 61440: $C_2^5.C_2^4.S_5$ x 3, $C_2^9.S_5$ x 2, $C_2^6.C_2^4.A_5$, $C_2^5.C_2^5.A_5$, $C_2^5.C_2^4.(D_5\times A_4)$, $C_2^3.C_2^8.D_{15}$, $C_2^3.C_2^8.C_{15}.C_2$, $C_2.C_2^8.(S_3\times F_5)$, $C_2^6.F_{16}:C_4$, $C_2^{10}.A_5$, $C_2^{10}.A_5$

Order 60480: $C_3\times A_8$

Order 57344: $C_2^{10}.F_8$, $C_2^9.F_8.C_2$

Order 55296: $Q_8:C_2^2.C_2^4.C_3^3.C_2^2$ x 2, $Q_8:C_2^2.A_4^2.D_6$ x 2, $C_2.C_2^8.C_3^3.C_4$ x 2, $C_2.C_2^8.C_3^3.C_2^2$ x 2, $Q_8:C_2^2.C_2^4.C_3^2.D_6$, $C_2^6.\He_3.D_4.C_2^2$, $C_2^3.C_2^6.C_3^2.D_6$, $C_2.C_2^8.C_3^2.D_6$, $C_2.C_2^6.A_4^2.C_3$, $C_2.C_2^6.(C_3\times A_4).D_6$, $S_4^3.C_2^2$

Order 51840: $\SO(5,3)$ x 3, $S_6\times \SOPlus(4,2)$, $C_2\times \SU(4,2)$

Order 49152: $C_2^7.C_2^4.C_6.C_2^2$ x 10, $C_2^5.C_2^4.D_6.C_2^3$ x 8, $C_2^7.C_2^4.C_3.D_4$ x 6, $C_2^5.C_2^6.C_6.C_2^2$ x 6, $C_2^6.C_2^6.C_6.C_2$ x 5, $C_2^5.C_2^4.C_2^3:D_6$ x 5, $C_2^7.C_2^4.C_{12}.C_2$ x 4, $C_2^5.C_2^6.D_6.C_2$ x 4, $C_2^5.C_2^4.C_2^2:D_{12}$ x 4, $C_2^5.C_2^4.(C_6.D_4).C_2$ x 4, $C_2.C_2^6.A_4.C_2^3.C_2^2$ x 4, $C_2^7.C_2^6.S_3$ x 3, $C_2^5.C_2^6.C_3.D_4$ x 3, $C_2^5.C_2^4.C_6.C_2^4$ x 3, $C_2^5.C_2^4.C_2^4:S_3$ x 3, $C_2.C_2^6.A_4.C_2^4.C_2$ x 3, $C_2^9.A_4.D_4$ x 2, $C_2^9.A_4.C_2^3$ x 2, $C_2^7.C_2^6.C_6$ x 2, $C_2^7.C_2^4.D_{12}$ x 2, $C_2^6.C_2^5.C_{12}.C_2$ x 2, $C_2^5.C_2^6.D_{12}$ x 2, $C_2^5.C_2^4.C_6.C_2^3.C_2$ x 2, $C_2.C_2^6.C_2^6.S_3$ x 2, $C_2^5.C_2^6:S_4$ x 2, $C_2^9.C_2^4.S_3$, $C_2^9.C_2^4.C_6$, $C_2^9.A_4.C_2^2.C_2$, $C_2^8.C_2^6.C_3$, $C_2^8.C_2^4.C_6.C_2$, $C_2^8.(D_4\times S_4)$, $C_2^7.A_4.C_2^4.C_2$, $C_2^6.C_2^6.D_6$, $C_2^6.C_2^5.C_6.C_2^2$, $C_2^6.C_2^4.D_6.C_2^2$, $C_2^5.C_2^5.C_6.C_2^3$, $C_2^5.C_2^4.C_6.D_4.C_2$, $C_2^4.C_2^6.D_6.C_2^2$, $C_2^3:A_4.C_2^4.C_2^5$, $C_2^3.C_2^6.C_6.C_2^3.C_2$, $C_2^3.C_2^6.A_4.D_4$, $C_2.C_2^6.S_4.C_2^4$, $C_2.C_2^6.C_2^6.C_6$

Order 46080: $C_2^6.S_6$ x 11, $C_2^6.S_6$ x 6, $S_4\times C_2^4:S_5$ x 4, $(C_2^4\times C_4):S_6$ x 4, $(C_2^4\times C_4):S_6$ x 4, $C_2^6.S_6$ x 4, $(D_4\times C_2^4):A_6$ x 4, $C_2^6.S_6$ x 4, $C_2^7.A_6$ x 3, $C_2^6.A_6.C_2$ x 2, $(C_2^5\times C_4).A_6$ x 2, $C_2^6.S_6$, $C_2^7:A_6$, $(A_4\times C_2^5):S_5$, $C_2^5:A_4.S_5$, $S_4\times C_2^5:A_5$, $A_4\times C_2^5:S_5$, $C_2^5.(S_4\times A_5)$, $C_2\wr S_5\times A_4$, $A_4.C_2\wr S_5$

Order 43200: $A_5:S_6$ x 2, $A_5\times S_6$ x 2, $S_5\times A_6$ x 2

Order 43008: $C_2^7.\PGL(2,7)$

Order 41472: $S_4\wr C_3$ x 2, $A_4^3:S_4$ x 2, $A_4^3:S_4$ x 2, $C_2\times A_4^3:A_4$, $C_2\times A_4^3:D_6$, $C_2^6:\Unitary(3,2)$

Order 40960: $C_2^5.C_2^4.C_2^2:F_5$ x 2, $C_2^5.C_2^4.F_5.C_2^2$, $C_2^5.C_2^4.D_5.C_2^3$, $C_2^5.C_2^4.C_5.C_4^2$, $C_2^5.C_2^4.C_4:F_5$, $C_2^5.C_2^4.(D_4\times D_5)$

Order 40320: $S_8$ x 5, $C_2\times A_8$ x 2

Order 36864: $C_2^5.A_4^2.D_4$ x 17, $C_2^5.A_4^2.C_2^3$ x 14, $C_2.C_2^6.A_4^2.C_2$ x 12, $C_2^5.C_2^6.C_3.C_6$ x 7, $C_2^5.C_2^4.(S_3\times A_4)$ x 7, $C_2^5.A_4^2.C_2^2.C_2$ x 7, $C_2^3.C_2^6.S_3^2.C_2$ x 5, $C_2.C_2^6.A_4.D_6.C_2$ x 5, $C_2^3:A_4.C_2^4.C_6.C_2^2$ x 4, $C_2^4.Q_8.A_4^2.C_2$ x 3, $C_2^3:A_4.C_2^5.C_6.C_2$ x 3, $C_2^3.C_2^6.C_3.D_6.C_2$ x 3, $C_2.C_2^6.D_6:S_4$ x 3, $Q_8.C_2^6.C_6^2.C_2$ x 2, $Q_8.C_2^6.A_4.S_3$ x 2, $C_2^7.A_4^2.C_2$ x 2, $C_2^7.A_4\wr C_2$ x 2, $C_2^5.C_2^6.C_3.S_3$ x 2, $C_2^5.C_2^4.C_6^2.C_2$ x 2, $C_2^5.A_4^2.C_4.C_2$ x 2, $C_2^3:A_4.C_2^6.C_6$ x 2, $C_2^3.C_2^6.A_4.C_6$ x 2, $C_2^9.\SOPlus(4,2)$ x 2, $C_2^9.\SOPlus(4,2)$ x 2, $Q_8:C_2^2.A_4^2.D_4$, $Q_8.C_2^6.S_3^2.C_2$, $Q_8.C_2^6.A_4.C_6$, $C_2^9.A_4.C_6$, $C_2^8.A_4^2$, $C_2^7.C_2^4.C_3.S_3$, $C_2^6.C_2^6.C_3^2$, $C_2^5.C_2^5.C_6^2$, $C_2^5.C_2^4.C_6.C_6.C_2$, $C_2^3.C_2^6.D_6:S_3$, $C_2^3.C_2^6.C_6.C_6.C_2$, $C_2^3.C_2^6.A_4.S_3$, $C_2^3.C_2^6.(S_3\times A_4)$, $C_2.C_2^6.D_6^2.C_2$, $C_2.C_2^6.C_3.C_2^4.C_6$, $C_2.C_2^6.(S_3\times S_4).C_2$, $C_2.C_2^6.(S_3\times A_4:C_4)$, $C_2.C_2^6.(C_3\times A_4).D_4$, $C_2^6.S_4^2$, $C_2^6.S_4^2$, $C_2^6.A_4^2:C_4$, $C_2^6.S_4^2$

Order 36288: $C_3.\SU(3,3).C_2$ x 2, $S_3\times \SU(3,3)$

Order 34560: $C_2\times S_4\times S_6$ x 3, $C_3\times C_2^5:A_6$

Order 32768: $C_2^5.C_2^6.C_2^4$ x 12, $C_2^7.C_2^5.C_2^3$ x 7, $C_2^7.C_2^4.C_2^4$ x 4, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^6.C_2^5.C_2^4$ x 2, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^3.C_2^4.C_2$, $C_2^5.C_2^5.C_2^5$, $C_2.C_2^6.C_2^5.C_2^3$

Order 32256: $C_2^3.C_2^6.C_{21}.C_3$, $C_3\times C_2^6:\GL(3,2)$

Order 31104: $S_3\wr S_4$

Order 30720: $C_2^7.C_2.S_5$ x 2, $C_2^7.F_{16}$, $C_2^5.F_{16}.C_4$, $C_2.C_2^8.C_{15}.C_4$, $C_2.C_2^8.C_5.D_6$, $C_2^5.C_2^4:A_5$, $C_2^5.(D_4\times S_5)$, $C_2^8.(D_5\times A_4)$, $C_2^9.A_5$

Order 30240: $S_3\times S_7$

Order 28800: $S_5\wr C_2$ x 2

Order 28672: $C_2^9.F_8$

Order 27648: $S_4^3:C_2$ x 8, $C_2^6:\He_3:\SD_{16}$ x 4, $Q_8:C_2^2.C_2^4.C_3^3.C_2$ x 3, $Q_8.C_2^6.C_3^3.C_2$ x 3, $C_2^3:A_4.A_4^2.C_2$ x 2, $Q_8:C_2^2.A_4^2.C_6$, $C_2^6:C_3.F_9.C_2$, $C_2^6.\He_3.Q_8.C_2$, $C_2.C_2^6.S_3^3$, $C_2.C_2^6.A_4.C_3.C_6$, $C_2\times S_4^3$, $C_2\times A_4^2:\GL(2,\mathbb{Z}/4)$, $C_2^7:\He_3:D_4$, $C_2\times A_4^3:D_4$, $C_2\times S_4^2:S_4$, $A_4^3.C_2^4$, $A_4^3.(C_2^2\times C_4)$, $C_2\times A_4^3:D_4$

Order 25920: $S_3^2\times S_6$ x 2, $S_3^2:S_6$ x 2, $\SU(4,2)$ x 2, $A_6:\SOPlus(4,2)$, $C_3^2:C_4\times S_6$, $A_6\times \SOPlus(4,2)$

Order 2448: $\PSL(2,17)$

Order 243: $C_3^4:C_3$

Order 25: $C_5^2$

Order 17: $C_{17}$

Order 7: $C_7$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 47377612800: $\Sp(8,2)$ ($C_1$)

Order 1: $C_1$ ($\Sp(8,2)$)

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 4 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $81 \times 81$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $80 \times 80$ rational character table.