Abstract group 1919877120.a: $C_2^5.\GL(5,2)\times S_3$

• Label: 1919877120.a
• Order: \( 2^{16} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 31 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 31 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $35$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_2^5.\GL(5,2)\times S_3$
Order:	\(1919877120\)\(\medspace = 2^{16} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 31 \)
Exponent:	\(26040\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 31 \)
Automorphism group:	$C_2^5.\GL(5,2)\times S_3$, of order \(1919877120\)\(\medspace = 2^{16} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 31 \)
Composition factors:	$C_2$ x 6, $C_3$, $\GL(5,2)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	21	24	28	30	31	42	62	84	93
Elements	1	238207	2904578	33985920	10665984	211676990	3809280	99993600	74661888	216980160	148561920	85327872	99041280	49996800	91422720	213319680	61931520	159989760	185794560	45711360	123863040	1919877120
Conjugacy classes	1	11	5	20	1	25	2	6	3	19	10	5	6	3	4	9	6	6	6	2	6	156
Divisions	1	11	5	20	1	25	1	6	3	19	5	3	3	3	2	5	1	3	1	1	1	120
Autjugacy classes	1	11	5	20	1	25	2	6	3	19	10	5	6	3	4	9	6	6	6	2	6	156

Dimension	1	2	30	31	60	62	124	155	217	248	280	310	315	434	465	496	560	620	630	651	868	930	960	992	1024	1085	1240	1302	1395	1736	1860	1890	1920	1984	2048	2170	2480	2604	2790	3255	3472	3720	3780	3968	4340	5580	6510	7440	9765	13020	19530
Irr. complex chars.	2	1	2	2	1	1	2	2	4	1	2	1	12	4	10	2	1	2	6	12	5	11	2	1	2	2	3	6	8	4	3	0	1	2	1	3	1	0	6	6	1	2	0	1	1	1	5	1	4	1	2	156
Irr. rational chars.	2	1	2	2	1	1	2	2	4	1	2	1	0	4	2	2	1	2	0	4	5	7	2	1	2	2	3	6	0	4	5	2	1	2	1	3	1	2	6	6	1	3	1	1	1	3	5	1	4	1	2	120

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	62	62	62
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $35$ $\langle(8,12)(10,15)(11,17)(13,19)(14,21)(16,23)(18,25)(22,28)(33,34), (1,2)(4,6) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^5.\GL(5,2))$ . $S_3$	$C_2^5$ . $(S_3.\GL(5,2))$	$S_3$ . $(C_2^5.\GL(5,2))$	$(D_6\times C_2^4)$ . $\GL(5,2)$	all 7
Aut. group:	$\Aut(D_6\times C_2^4)$

Elements of the group are displayed as permutations of degree 35.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 9 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:	not computed
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:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
31-Sylow subgroup:	$P_{ 31 } \simeq$ $C_{31}$

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 12288 (order at least 156240) are shown, as well as all normal subgroups of any index.

Order 1919877120: $C_2^5.\GL(5,2)\times S_3$

Order 959938560: $(C_2^4\times C_6).\GL(5,2)$

Order 639959040: $C_2^6.\GL(5,2)$

Order 319979520: $C_2^5.\GL(5,2)$

Order 61931520: $(C_2^7\times C_6).C_2^2.A_8$, $S_3\times C_2^5.C_2^4.A_8$

Order 59996160: $S_3.\GL(5,2)$

Order 30965760: $(C_2^4\times C_6).C_2^4.A_8$ x 2, $(C_2^7\times C_6).C_2.A_8$, $(C_2^4\times S_3).C_2^4.A_8$

Order 29998080: $C_3\times \GL(5,2)$

Order 20643840: $C_2^9.C_2.A_8$, $C_2^5.C_2^5.A_8$

Order 19998720: $C_2\times \GL(5,2)$

Order 15482880: $(C_2^7\times C_6).A_8$

Order 12386304: $C_2^9.(S_3\times S_4).\PSL(2,7)$, $C_2^8.S_3^2.C_2^3.\PSL(2,7)$

Order 10321920: $C_2^9.A_8$, $C_2^8.C_2.A_8$, $C_2^5.C_2^4.A_8$, $C_2^5.C_2^4.A_8$

Order 9999360: $\GL(5,2)$

Order 7741440: $(C_2^7\times C_6).C_2^2.A_7$, $(C_2^5\times S_3).C_2^4.A_7$

Order 6193152: $C_2^8.C_3^2.C_2^4.\PSL(2,7)$ x 2, $C_2^9.C_3:S_4.\PSL(2,7)$, $C_2^9.A_4.C_6.\PSL(2,7)$, $(C_2^8\times C_6).A_4.C_2.\PSL(2,7)$, $(C_2^7\times C_6).C_3.C_2^4.\PSL(2,7)$

Order 5160960: $C_2^8.A_8$

Order 4128768: $S_3\times C_2^5.C_2^7:\GL(3,2)$ x 2, $C_2^9.A_4.C_2^2.\PSL(2,7)$, $(C_2^6\times C_6).C_2^6.\PSL(2,7)$, $C_2^6.C_2^6:(S_3\times \GL(3,2))$

Order 3870720: $(C_2^7\times C_6).C_2.A_7$ x 2, $(C_2^5\times S_3).A_8$ x 2, $(C_2^4\times S_3).C_2^4.A_7$, $(C_2^4\times C_6).C_2^4.A_7$

Order 3096576: $C_2^8.C_3^2.C_2^3.\PSL(2,7)$, $C_2^6.C_3^2.C_2^5.\PSL(2,7)$, $C_2^5.C_2^6.C_3^2.\PSL(2,7)$

Order 2580480: $C_2^9.C_2.A_7$, $C_2^6.C_2^4.A_7$

Order 2211840: $(C_2^7\times C_6).C_2^2.A_6.C_2$, $(C_2^5\times S_3).C_2^4.A_6.C_2$

Order 2064384: $(C_2^6\times C_6).C_2^5.\PSL(2,7)$ x 5, $(C_2^7\times C_6).C_2^4.\PSL(2,7)$ x 3, $C_2^5.C_2^6:(S_3\times \GL(3,2))$ x 3, $C_2^9.S_4.\PSL(2,7)$ x 2, $C_2^8.(S_3\times C_2^3:\GL(3,2))$ x 2, $C_2^9.A_4.C_2.\PSL(2,7)$, $C_2^8.C_6.C_2^3.\PSL(2,7)$, $C_2^5.C_2^6.S_3.\PSL(2,7)$, $(C_2^5\times C_6).C_2^6.\PSL(2,7)$, $(C_2^4\times S_3).C_2^6.C_2.\PSL(2,7)$, $(C_2^3\times C_3:D_4).C_2^6.\PSL(2,7)$

Order 1935360: $C_2^4.A_8\times S_3$ x 5, $(C_2^4\times C_6).A_8$ x 2, $(C_2^7\times C_6).A_7$

Order 1769472: $C_2^7.C_2^6.S_3^3$, $C_2^6.C_2^6.S_3^3.C_2$, $C_2.C_2^6.C_2^6.S_3^3$

Order 1548288: $C_2^8.S_3^2.\PSL(2,7)$ x 2, $C_2^6.C_3^2.C_2^4.\PSL(2,7)$ x 2, $C_3.C_2^6:C_3.C_2^4.\PSL(2,7)$, $C_2^8.F_8.C_3^3.C_2^2$, $C_2^5.C_2^6.C_{21}.(C_6\times S_3)$

Order 1376256: $C_2^6.C_2^7:\GL(3,2)$ x 2, $C_2^8.C_2^5.\PSL(2,7)$

Order 1290240: $C_2^8.C_2.A_7$ x 2, $C_2^6.A_8$ x 2, $C_2^9.A_7$, $C_2^5.C_2^4.A_7$

Order 1105920: $(C_2^7\times C_6).C_2.A_6.C_2$ x 3, $(C_2^4\times C_6).C_2^4.A_6.C_2$ x 3, $(C_2^4\times S_3).C_2^4.A_6.C_2$ x 2, $(C_2^7\times C_6).C_6.C_2.S_5$, $(C_2^7\times C_6).C_2^2.A_6$, $(C_2^5\times S_3).C_2^4.A_6$, $(C_2^4\times C_6).C_2^4.C_6.S_5$

Order 1032192: $(C_2^8\times C_6).C_2^2.\PSL(2,7)$ x 4, $(C_2^6\times C_6).C_2^4.\PSL(2,7)$ x 3, $C_2^9.A_4.\PSL(2,7)$ x 2, $C_2^8.C_3.C_2^3.\PSL(2,7)$ x 2, $(C_2^4\times C_6).C_2^6.\PSL(2,7)$ x 2, $(C_2^5\times C_6).C_2^5.\PSL(2,7)$, $(C_2^3\times C_6).C_2^6.C_2.\PSL(2,7)$, $C_2^9.(D_6\times \GL(3,2))$

Order 967680: $(C_2^3\times C_6).A_8$ x 3

Order 884736: $C_2^8.C_2^4.S_3^3$ x 7, $C_2^8.C_2^4.C_3^3.C_2^3$ x 6, $C_2^9.C_2^4.C_3^2.D_6$ x 3, $C_2^7.C_2^6.C_3^2.D_6$ x 2, $C_2^5.C_2^6.(A_4\times S_3^2)$ x 2, $C_2^9.A_4^2.D_6$, $C_2^8.C_6^2.A_4.C_2^3$, $C_2^8.C_2.C_6^2.A_4.C_2^2$, $C_2^7.C_2^6.C_3^3.C_2^2$, $C_2^7.C_2^4.(A_4\times S_3^2)$, $C_2^6.C_2^6.S_3^3$, $C_2^5.C_2^6.C_6^2.D_6$, $C_2^5.C_2^6.(C_3\times S_3\times S_4)$, $C_2.C_2^6.C_2^6.C_3^2.D_6$

Order 774144: $C_2^8.F_8.C_3^3.C_2$ x 3, $C_2^8.C_3:S_3.\PSL(2,7)$ x 2, $C_2^8.C_3.C_6.\PSL(2,7)$ x 2, $(C_2^7\times C_6).C_6.\PSL(2,7)$ x 2, $C_2^9.C_7.C_6^2.C_6$, $C_2^9.(C_{21}\times A_4).C_6$, $C_2^6.C_3^2.C_2^3.\PSL(2,7)$, $C_2^5.C_2^6.C_{21}.C_3.C_6$

Order 737280: $C_2^8.C_2^2.A_6.C_2$, $C_2^5.C_2^5.A_6.C_2$

Order 688128: $C_2^7.C_2^5.\PSL(2,7)$ x 4, $C_2^5.C_2^7:\GL(3,2)$ x 4, $C_2^5.C_2^6.C_2.\PSL(2,7)$ x 2, $C_2^9.C_2^3.\PSL(2,7)$, $C_2^8.C_2^4.\PSL(2,7)$, $C_2^6.C_2^6.\PSL(2,7)$

Order 645120: $C_2^5.A_8$ x 7, $C_2^8.A_7$

Order 589824: $C_2^8.C_2^4.D_6^2$ x 2, $C_2^9.A_4^2.C_2^3$, $C_2^9.A_4.D_6.C_2^3$, $C_2^8.C_6^2.C_2^5.C_2$, $C_2^5.C_2^6.D_6^2.C_2$, $C_2^5.C_2^6.(C_2\times S_3\times S_4)$

Order 552960: $(C_2^7\times C_6).C_6.S_5$ x 4, $(C_2^7\times C_6).A_6.C_2$ x 4, $C_2^5.C_2^4.C_3^2.S_5$ x 2, $(C_2^7\times C_6).C_2.A_6$ x 2, $C_2^8.C_3^2.C_2.S_5$, $C_2^8.C_3.C_6.S_5$, $(C_2^7\times C_6).C_6.C_2.A_5$, $(C_2^4\times S_3).C_2^4.A_6$, $(C_2^4\times C_6).C_2^4.C_6.A_5$, $(C_2^4\times C_6).C_2^4.A_6$

Order 516096: $(C_2^8\times C_6).C_2.\PSL(2,7)$ x 3, $(C_2^7\times C_6).C_2^2.\PSL(2,7)$ x 3, $C_2^9.(S_3\times \GL(3,2))$ x 3, $C_2^8.F_8.(C_6\times S_3)$ x 2, $C_2^5.C_2^6.C_{21}.C_6.C_2$ x 2, $S_3.C_2^6.C_2^3.\PSL(2,7)$, $C_6.C_2^6.C_2^3.\PSL(2,7)$, $C_2^9.C_{42}.C_6.C_2^2$, $C_2^9.A_4.C_{21}.C_2^2$, $C_2^8.D_6.\PSL(2,7)$, $C_2^8.C_3^2.F_8.C_2^2$, $(C_2^6\times C_6).C_2^3.\PSL(2,7)$, $(C_2^5\times S_3).C_2^4.\PSL(2,7)$, $(C_2^5\times C_6).C_2^4.\PSL(2,7)$, $(C_2^4\times S_3).C_2^5.\PSL(2,7)$, $(C_2^4\times C_6).C_2^5.\PSL(2,7)$, $C_2^7.(S_4\times \GL(3,2))$, $C_2^9.(C_6\times \GL(3,2))$

Order 483840: $(C_2^5\times S_3).A_7$ x 2

Order 442368: $C_2^8.C_2^4.C_3^2.D_6$ x 10, $C_2^8.C_2^4.C_3^3.C_2^2$ x 9, $C_2^6.C_2^6.C_3^2.D_6$ x 7, $C_2^8.C_6^2.A_4.C_2^2$ x 5, $C_2^5.C_2^6.C_6^2.C_6$ x 3, $C_2^8.C_2^2.S_3^3.C_2$ x 2, $C_2^9.(S_4\times S_3^2)$ x 2, $C_2^9.C_2^4.C_3^3.C_2$, $C_2^9.A_4^2.C_6$, $C_2^9.A_4.(C_6\times A_4)$, $C_2^9.(C_3\times A_4^2).C_2$, $C_2^8.C_6^2.C_6.C_2^3$, $C_2^8.C_6.C_2^4.C_3.C_6$, $C_2^8.C_3.C_2^4.C_6^2$, $C_2^8.C_2^4.C_3.C_6^2$, $C_2^8.C_2.C_6^2.A_4.C_2$, $C_2^7.C_2^4.C_6^2.C_6$, $C_2^6.C_2^6.C_3^3.C_2^2$, $C_2^5.C_2^6.C_3^2:S_4$, $C_2^5.C_2^4.A_4^2.C_6$, $(C_2^8\times C_6).A_4^2.C_2$, $(C_2^7\times C_6).C_2^4.C_6^2$, $(C_2^6\times C_6).C_2^6.C_3.C_6$

Order 387072: $C_2^6.S_3^2.\PSL(2,7)$ x 3, $C_2^8.C_3^2.\PSL(2,7)$ x 2, $C_2^9.C_{21}.(C_6\times S_3)$, $C_2^9.C_7.C_6^2.C_3$, $C_2^8.F_8.C_3^3$

Order 368640: $C_2^8.C_2.A_6.C_2$ x 3, $C_2^5.C_2^4.A_6.C_2$ x 3, $C_2^9.A_6.C_2$ x 2, $(C_2^5\times S_3).C_2^4.S_5$ x 2, $C_2^9.C_2.A_6$, $C_2^8.C_6.C_2.S_5$, $C_2^6.C_2^4.A_6$, $C_2^5.C_2^4.C_6.S_5$, $(C_2^4\times S_3).C_2^5.S_5$, $(C_2^4\times C_6).C_2^5.S_5$

Order 344064: $C_2^7.C_2^4.\PSL(2,7)$ x 5, $C_2^9.C_2^2.\PSL(2,7)$ x 2, $C_2^5.C_2^6.\PSL(2,7)$ x 2, $C_2^{10}.C_2.\PSL(2,7)$, $C_2^8.C_2^3.\PSL(2,7)$

Order 322560: $C_2^4.A_8$ x 3

Order 294912: $C_2^8.C_6^2.C_2^4.C_2$ x 19, $C_2^8.C_2^4.S_3^2.C_2$ x 11, $C_2^8.C_6^2.C_2.C_2^4$ x 6, $C_2^9.A_4^2.C_2^2$ x 5, $C_2^9.(C_6\times A_4).C_2^3$ x 5, $C_2^7.C_2^6.S_3^2$ x 5, $C_2^8.D_6^2.C_2^3$ x 4, $C_2^8.C_2^4.C_3.D_6.C_2$ x 4, $C_2^8.A_4^2.C_2^3$ x 4, $C_2^6.C_2^6.S_3^2.C_2$ x 3, $C_2^5.C_2^6.C_3.D_6.C_2^2$ x 3, $C_2.C_2^6.C_2^6.S_3^2$ x 3, $C_2^9.C_2^2.D_6:D_6$ x 2, $C_2^9.A_4.D_6.C_2^2$ x 2, $C_2^9.A_4.C_6.C_2^3$ x 2, $C_2^8.C_2^4.C_3^2.C_2^3$ x 2, $C_2^7.C_2^6.C_3.D_6$ x 2, $C_2^6.C_2^6.C_3^2.C_2^3$ x 2, $C_2^5.C_2^6.D_6^2$ x 2, $C_2^5.C_2^6.D_6:D_6$ x 2, $C_2^5.C_2^6.C_3.C_6.C_2^3$ x 2, $C_2^5.C_2^6.A_4.D_6$ x 2, $C_2^5.C_2^6.(S_3\times D_{12})$ x 2, $C_2^5.C_2^6.(C_4\times S_3^2)$ x 2, $C_2^8.C_2^5:S_3^2$ x 2, $C_2^9.C_2^2.D_6^2$, $C_2^8.C_6.C_2^4.C_6.C_2$, $C_2^8.C_6.C_2.C_6.C_2^4$, $C_2^8.C_3.C_2^4.C_6.C_2^2$, $C_2^7.C_2^4.D_6^2$, $C_2^7.C_2^4.C_6^2.C_2^2$, $C_2^7.C_2^4.C_3.C_6.C_2^3$, $C_2^6.C_2^6.C_3.D_6.C_2$, $C_2^6.C_2^4.C_3.C_6.C_2^4$, $C_2^5.C_2^6.C_4:S_3^2$, $(C_2^7\times C_6).C_2^4.C_6.C_2^2$, $(C_2^6\times C_6).C_2^4.C_6.C_2^3$, $(C_2^4\times C_6).C_2^6.C_6.C_2^3$, $(C_2^4\times C_6).C_2^5.A_4.C_2^3$, $C_2^9.C_2^4:S_3^2$, $S_3\times C_2^5.C_2^6:S_4$

Order 276480: $C_2^8.C_3^2.S_5$ x 4, $(C_2^7\times C_6).C_6.A_5$ x 2, $C_2^8.C_3^2.C_2.A_5$, $C_2^5.C_2^4.C_3^2.A_5$, $(C_2^7\times C_6).A_6$

Order 258048: $(C_2^4\times C_6).C_2^4.\PSL(2,7)$ x 11, $C_2^5.C_2^6.C_{21}.C_6$ x 9, $(C_2^4\times S_3).C_2^4.\PSL(2,7)$ x 9, $C_2^8.F_8.C_3.C_6$ x 6, $(C_2^7\times C_6).C_2.\PSL(2,7)$ x 6, $(C_2^6\times C_6).C_2^2.\PSL(2,7)$ x 6, $C_2^9.C_{42}.C_6.C_2$ x 3, $C_2^9.C_{21}.C_6.C_2^2$ x 3, $C_2^8.S_3.\PSL(2,7)$ x 3, $(C_2^5\times S_3).C_2^3.\PSL(2,7)$ x 3, $D_6.C_2^7:\GL(3,2)$ x 3, $C_2^6.(S_4\times \GL(3,2))$ x 3, $C_2^9.(C_7\times A_4).C_6$ x 2, $(C_2^5\times C_6).C_2^3.\PSL(2,7)$ x 2, $C_2^3\wr C_3.\GL(3,2)$ x 2, $S_3.C_2^6.C_2^2.\PSL(2,7)$, $D_{12}.C_2^6.\PSL(2,7)$, $C_2^9.C_{21}.A_4.C_2$, $C_2^9.C_{14}.C_6^2$, $C_2^9.C_7.(C_6\times A_4)$, $C_2^9.(C_7\times C_3:S_4)$, $C_2^8.C_6.\PSL(2,7)$, $C_2^8.C_3^2.F_8.C_2$, $C_2^8.C_3.F_8.C_6$, $(C_2^8\times C_6).\PSL(2,7)$, $(C_2^7\times C_6).F_8.C_6$, $(C_2^4\times C_6).C_2^6.C_{42}$, $C_2^8.(C_6\times \GL(3,2))$

Order 241920: $(C_2^4\times S_3).A_7$ x 5, $(C_2^4\times C_6).A_7$ x 2, $D_6\times A_8$

Order 221184: $C_2^8.C_2^2.S_3^3$ x 18, $C_2^6.C_2^4.S_3^3$ x 7, $C_2^8.C_2^4.C_3^3.C_2$ x 6, $C_2^5.C_2^6.C_3^2.D_6$ x 5, $C_2^8.C_6^2.D_6.C_2$ x 4, $C_2^8.(A_4\times S_3^2).C_2$ x 3, $C_2^6.C_2^4.C_3^3.C_2^3$ x 3, $C_2^6.A_4^2.D_6.C_2$ x 3, $C_2^5.C_2^6.C_3^3.C_2^2$ x 3, $C_2^9.C_6^2.D_6$ x 2, $C_2^8.C_6^2.C_6.C_2^2$ x 2, $C_2^8.C_6^2.A_4.C_2$ x 2, $C_2^8.C_3.C_2^4.C_3.C_6$ x 2, $C_2^7.C_6^2.A_4.C_2^2$ x 2, $C_2^6.C_2^6.C_3^3.C_2$ x 2, $C_2^6.A_4^2.C_6.C_2^2$ x 2, $C_2^9.C_2^4.C_3^3$, $C_2^9.(C_3\times A_4).D_6$, $C_2^8.S_3^3.C_2^2$, $C_2^8.C_2.(A_4\times S_3^2)$, $C_2^8.A_4^2.C_6$, $C_2^8.A_4.S_3^2.C_2$, $C_2^8.A_4.C_6^2.C_2$, $C_2^7.C_2^4.C_3^2.D_6$, $C_2^6.C_6^2.A_4.C_2^3$, $C_2^6.A_4.C_3.A_4.C_2^3$, $C_2^6.(C_3\times A_4^2).C_2^3$, $C_2^5.C_2^4.S_3^3.C_2$, $C_2^4.C_2^6.S_3^3$, $C_2.C_2^6.C_2^6.C_3^3$, $C_2.C_2^6.A_4^2.C_6.C_2$, $(C_2^7\times C_6).C_2^4.C_3.C_6$, $(C_2^5\times C_6).C_2^6.C_3.C_6$

Order 196608: $C_3:(C_2^7.C_2^5.C_2^4)$, $C_2^8.C_6.C_2^5.C_2^2$, $C_2^8.C_2^4.C_6.C_2^3$, $C_2^7.C_2^4.C_6.C_2^4$, $C_2^5.C_2^6.C_6.C_2^4$

Order 193536: $C_2^9.C_{21}.C_3.C_6$ x 3, $C_2^6:C_3.C_6.\PSL(2,7)$ x 3, $C_2^6.C_3:S_3.\PSL(2,7)$ x 3, $(C_2^5\times C_6).C_6.\PSL(2,7)$ x 3, $C_2^8.C_{21}.(C_6\times S_3)$, $C_2^5.(S_3^2\times \GL(3,2))$

Order 184320: $(C_2^4\times C_6).C_2^4.S_5$ x 7, $(C_2^4\times S_3).C_2^4.S_5$ x 6, $C_2^8.C_6:S_5$ x 5, $C_2^8.A_6.C_2$ x 4, $C_2^5.C_2^4.C_3.S_5$ x 3, $(C_2^7\times C_6).C_2.S_5$ x 3, $C_2^8.C_6.S_5$ x 2, $C_2^8.C_2.A_6$ x 2, $(C_2^7\times C_6).C_2^2.A_5$ x 2, $(C_2^5\times S_3).C_2^4.A_5$ x 2, $C_2^9.A_6$, $C_2^8.C_{15}.C_6.C_2^3$, $C_2^8.C_6.C_2.A_5$, $C_2^5.C_2^4.C_6.A_5$, $C_2^5.C_2^4.A_6$, $C_2^5.C_2^4.(S_3\times C_3:F_5)$

Order 172032: $C_2^9.C_2.\PSL(2,7)$ x 6, $C_2^8.C_2^2.\PSL(2,7)$ x 2, $C_2^6.C_2^4.\PSL(2,7)$ x 2, $(C_2\times C_6).C_2^6.F_8.C_2^2$ x 2, $C_2^{10}.\PSL(2,7)$, $C_2^9.C_7.C_6.C_2^3$, $C_2^8.F_8.C_6.C_2$, $C_2^8.(C_6\times F_8).C_2$, $C_2^5.C_2^6.C_{42}.C_2$, $C_2^5.C_2^6.C_7.C_6.C_2$, $C_2^5.C_2^5.\PSL(2,7)$, $C_2^4.C_2^6.\PSL(2,7)$, $(C_2^6\times C_6).F_8.C_2^3$

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

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

Order 29760: $C_2^5.C_{93}.C_{10}$

Order 192: $D_6\times C_2^4$

Order 96: $C_2^4\times C_6$

Order 32: $C_2^5$

Order 31: $C_{31}$

Order 27: $C_3^3$

Order 7: $C_7$

Order 6: $S_3$

Order 5: $C_5$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1919877120: $C_2^5.\GL(5,2)\times S_3$ ($C_1$)

Order 959938560: $(C_2^4\times C_6).\GL(5,2)$ ($C_2$)

Order 319979520: $C_2^5.\GL(5,2)$ ($S_3$)

Order 192: $D_6\times C_2^4$ ($\GL(5,2)$)

Order 96: $C_2^4\times C_6$ ($C_2\times \GL(5,2)$)

Order 32: $C_2^5$ ($S_3.\GL(5,2)$)

Order 6: $S_3$ ($C_2^5.\GL(5,2)$)

Order 3: $C_3$ ($C_2^6.\GL(5,2)$)

Order 1: $C_1$ ($C_2^5.\GL(5,2)\times S_3$)

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

Character theory

Complex character table

See the $156 \times 156$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $120 \times 120$ rational character table.