Abstract group 26880.d: $C_2^3:\GL(3,2)\times D_{10}$

• Label: 26880.d
• Order: \( 2^{8} \cdot 3 \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $70$
• Rank: $2$

Group information

Description:	$C_2^3:\GL(3,2)\times D_{10}$
Order:	\(26880\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2\times C_2^4.\PSL(2,7)\times F_5$, of order \(107520\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_5$, $\PSL(2,7)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	10	14	15	20	30	35	70
Elements	1	1103	224	5040	4	5152	384	732	4224	896	3360	2688	1536	1536	26880
Conjugacy classes	1	15	1	12	2	7	2	14	6	2	12	6	4	4	88
Divisions	1	15	1	12	1	7	1	7	3	1	6	3	1	1	60
Autjugacy classes	1	8	1	6	1	5	2	5	4	1	4	3	2	2	45

Dimension	1	2	3	4	6	7	8	12	14	16	21	24	28	32	42	56	84
Irr. complex chars.	4	4	8	0	12	12	4	4	16	4	8	0	4	0	8	0	0	88
Irr. rational chars.	4	0	0	2	8	12	4	0	4	0	8	4	6	2	0	2	4	60

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$70$
Rank:	$2$
Inequivalent generating pairs:	$1026$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	14	28
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(3,5)(4,7)(8,13)(9,14)(10,15)(11,12), (1,2)(4,5)(6,7), (1,2)(3,5)(4,7)(8,14,9,13) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $D_5$ $\, \times\, $ $(C_2^3:\GL(3,2))$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $(D_5\times \GL(3,2))$	$C_5$ $\,\rtimes\,$ $(C_2^5:\GL(3,2))$	$C_{10}$ $\,\rtimes\,$ $(C_2^4:\GL(3,2))$	$(C_2^3\times D_{10})$ $\,\rtimes\,$ $\PSL(2,7)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 486754 subgroups in 4270 conjugacy classes, 21 normal (15 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_5\times C_2^3:\GL(3,2)$
Commutator:	$G' \simeq$ $C_5\times C_2^3:\GL(3,2)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^3:\GL(3,2)\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_2\times \GL(3,2)$
Radical:	$R \simeq$ $C_2^3\times D_{10}$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times \GL(3,2)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 26880: $C_2^3:\GL(3,2)\times D_{10}$

Order 13440: $D_5\times C_2^3:\GL(3,2)$ x 2, $C_2^3:\GL(3,2)\times C_{10}$

Order 6720: $C_5\times C_2^3:\GL(3,2)$

Order 5376: $C_2^5:\GL(3,2)$

Order 3840: $C_2^3:S_4\times D_{10}$, $C_2^3:S_4\times D_{10}$

Order 3360: $D_{10}\times \PSL(2,7)$ x 2, $D_5\times F_8:C_6$

Order 2688: $C_2^4:\GL(3,2)$ x 3

Order 1920: $D_5\times C_2^3:S_4$ x 8, $D_5\times C_2^3:S_4$ x 4, $C_2^2:S_4\times D_{10}$ x 2, $(C_2^2\times D_{10}):S_4$ x 2, $Q_8:A_4\times D_{10}$, $(C_2^3\times C_{10}):S_4$, $C_2^3:A_4\times D_{10}$, $C_2^5:D_{30}$, $C_2^3:S_4\times C_{10}$, $C_2^3:S_4\times C_{10}$

Order 1680: $D_5\times \GL(3,2)$ x 4, $D_5\times F_8:C_3$ x 2, $C_{10}\times \GL(3,2)$ x 2, $F_8:C_{30}$

Order 1344: $C_2^3:\GL(3,2)$

Order 1280: $C_2\wr C_2^2\times D_{10}$

Order 1120: $D_{10}\times F_8$

Order 960: $(C_2\times D_{10}):S_4$ x 8, $D_5\times C_2^2:S_4$ x 8, $D_5\times C_2^3:A_4$ x 4, $C_2^4:D_{30}$ x 4, $C_5\times C_2^3:S_4$ x 4, $C_2\times D_{10}\times S_4$ x 3, $C_2^4:D_{30}$ x 2, $C_2^2:S_4\times C_{10}$ x 2, $D_5\times Q_8:A_4$ x 2, $(C_2^2\times C_{10}):S_4$ x 2, $C_5\times C_2^3:S_4$ x 2, $C_2^5:C_{30}$, $(C_2^2\times D_{10}):A_4$, $C_2^2:A_4\times D_{10}$, $C_{10}\times Q_8:A_4$

Order 840: $C_5\times \PSL(2,7)$ x 2, $F_8:C_{15}$

Order 768: $C_2^5:S_4$, $C_2^5:S_4$

Order 672: $C_2^2\times \GL(3,2)$ x 2, $C_2\times F_8:C_6$

Order 640: $C_2\wr C_2^2\times D_5$ x 16, $C_2^5:D_{10}$ x 3, $C_2^5:D_{10}$ x 3, $C_2^3:C_4\times D_{10}$ x 3, $C_2^4:D_{20}$ x 3, $D_{20}:C_2^4$, $C_2\wr C_2^2\times C_{10}$, $(D_4\times C_{10}):D_4$

Order 560: $D_5\times F_8$ x 2, $C_{10}\times F_8$

Order 480: $D_{10}\times S_4$ x 38, $C_2^4:D_{15}$ x 4, $C_5\times C_2^2:S_4$ x 4, $C_2\times A_4\times D_{10}$ x 3, $C_2^3:D_{30}$ x 3, $C_2\times C_{10}\times S_4$ x 3, $C_2^4:C_{30}$ x 2, $(C_2\times D_{10}):A_4$ x 2, $D_5\times C_2^2:A_4$ x 2, $C_2^4:C_{30}$, $C_5\times Q_8:A_4$, $D_{10}\times \SL(2,3)$

Order 420: $C_{70}:C_6$

Order 384: $C_2^4:S_4$ x 12, $C_2^4:S_4$ x 6, $C_2^2\wr S_3$ x 2, $C_2^5:A_4$, $C_2^5:A_4$

Order 336: $C_2\times \GL(3,2)$ x 6, $F_8:C_6$ x 3

Order 320: $C_2^4:D_{10}$ x 32, $C_2^4:D_{10}$ x 24, $C_2^3:C_4\times D_5$ x 24, $C_2^3:D_{20}$ x 24, $D_{20}:C_2^3$ x 10, $C_2\wr C_2^2\times C_5$ x 8, $(C_2\times C_{20}):D_4$ x 8, $C_2^3.D_{20}$ x 6, $C_{20}:C_2^4$ x 5, $C_2^4:D_{10}$ x 5, $C_2^3:D_{20}$ x 5, $C_2^4.D_{10}$ x 5, $C_2^4:C_{20}$ x 3, $C_2^4.D_{10}$ x 3, $C_2^4:D_{10}$ x 3, $C_2^5:C_{10}$ x 3, $C_2^4:D_{10}$ x 3, $C_2^4\times D_{10}$, $C_{10}.C_2^5$, $D_{20}:C_2^3$, $D_{20}:C_2^3$

Order 280: $C_5\times F_8$

Order 256: $C_2^5:D_4$

Order 240: $D_5\times S_4$ x 56, $A_4\times D_{10}$ x 20, $C_{10}:S_4$ x 20, $C_{10}\times S_4$ x 20, $C_2^3:C_{30}$ x 3, $D_5\times \SL(2,3)$ x 2, $C_2^4:C_{15}$, $D_6\times D_{10}$, $C_{10}\times \SL(2,3)$

Order 224: $C_2^2\times F_8$

Order 210: $C_{35}:C_6$ x 2, $C_7:C_{30}$

Order 192: $C_2^3:S_4$ x 16, $C_2^3:S_4$ x 12, $C_2^4:A_4$ x 6, $C_2^3:S_4$ x 4, $C_2^3\times S_4$ x 3, $C_2^4:A_4$ x 3, $C_2^2\wr C_3$

Order 168: $\PSL(2,7)$ x 2, $F_8:C_3$

Order 160: $D_4\times D_{10}$ x 77, $D_{10}:D_4$ x 34, $C_2^2:D_{20}$ x 34, $D_{10}.D_4$ x 34, $D_4:D_{10}$ x 30, $C_2^2.D_{20}$ x 24, $C_2^3\times D_{10}$ x 17, $D_4:D_{10}$ x 16, $D_4:D_{10}$ x 16, $C_2^4:C_{10}$ x 16, $C_2^4:D_5$ x 16, $C_2^3:C_{20}$ x 12, $C_2^3.D_{10}$ x 12, $C_2^3:D_{10}$ x 10, $C_2^2.D_{20}$ x 7, $C_{10}.C_2^4$ x 5, $C_{20}:C_2^3$ x 5, $C_2^2\times D_{20}$ x 5, $C_{20}:C_2^3$ x 5, $C_2^3:C_{20}$ x 5, $C_2^3.D_{10}$ x 5, $D_4:D_{10}$ x 3, $D_{20}:C_2^2$ x 3, $C_2^4\times C_{10}$, $C_{10}.C_2^4$, $Q_8:D_{10}$, $Q_8\times D_{10}$

Order 140: $C_7\times D_{10}$

Order 128: $C_2^4:D_4$ x 24, $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 3, $D_4:C_2^4$

Order 120: $D_5\times A_4$ x 16, $C_5:S_4$ x 16, $C_5\times S_4$ x 16, $S_3\times D_{10}$ x 12, $C_{10}\times A_4$ x 11, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_5\times \SL(2,3)$

Order 112: $C_2\times F_8$ x 3

Order 105: $C_7:C_{15}$

Order 96: $C_2^2\times S_4$ x 44, $C_2^2:S_4$ x 8, $C_2^3:A_4$ x 4, $C_2^3:A_4$ x 3, $C_2^3\times A_4$ x 3, $Q_8:A_4$, $C_2^2\times \SL(2,3)$

Order 84: $C_{14}:C_6$

Order 80: $D_4\times D_5$ x 136, $C_2^2\times D_{10}$ x 106, $C_{10}:D_4$ x 74, $C_4\times D_{10}$ x 38, $D_4\times C_{10}$ x 37, $C_2\times D_{20}$ x 37, $D_{10}:C_4$ x 28, $D_4:D_5$ x 24, $D_{20}:C_2$ x 24, $C_2^2:C_{20}$ x 17, $C_{10}.D_4$ x 17, $C_2^3\times C_{10}$ x 10, $D_4:C_{10}$ x 8, $D_{20}:C_2$ x 8, $C_2^2\times C_{20}$ x 5, $C_2^2.D_{10}$ x 5, $Q_8\times D_5$ x 4, $Q_8\times C_{10}$, $C_{10}:Q_8$

Order 70: $C_7\times D_5$ x 2, $C_{70}$

Order 64: $C_2\wr C_2^2$ x 64, $C_2^3:D_4$ x 48, $C_2^4:C_4$ x 36, $D_4:C_2^3$ x 15, $D_4\times C_2^3$ x 5, $C_2^4:C_4$ x 5, $C_2^6$, $D_4:C_2^3$

Order 60: $S_3\times D_5$ x 16, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_3\times D_{10}$ x 6, $C_5\times A_4$ x 5, $C_2\times C_{30}$

Order 56: $F_8$

Order 48: $C_2\times S_4$ x 96, $C_2^2\times A_4$ x 23, $C_2\times \SL(2,3)$ x 3, $C_2^2\times D_6$, $C_2^2:A_4$

Order 42: $C_7:C_6$ x 3

Order 40: $C_2\times D_{10}$ x 181, $C_5:D_4$ x 56, $C_2^2\times C_{10}$ x 35, $C_4\times D_5$ x 32, $D_{20}$ x 28, $C_5\times D_4$ x 28, $C_2\times C_{20}$ x 18, $C_{10}:C_4$ x 18, $C_5:Q_8$ x 4, $C_5\times Q_8$ x 2

Order 35: $C_{35}$

Order 32: $C_2^2\wr C_2$ x 100, $C_2^2\times D_4$ x 97, $D_4:C_2^2$ x 51, $C_2^3:C_4$ x 51, $C_2^3:C_4$ x 48, $D_4:C_2^2$ x 24, $C_2^5$ x 18, $C_2^3\times C_4$ x 5, $C_2^2\times Q_8$

Order 30: $D_{15}$ x 4, $C_3\times D_5$ x 4, $C_5\times S_3$ x 4, $C_{30}$ x 3

Order 28: $C_2\times C_{14}$

Order 24: $S_4$ x 32, $C_2\times A_4$ x 27, $C_2\times D_6$ x 14, $\SL(2,3)$, $C_2^2\times C_6$

Order 21: $C_7:C_3$

Order 20: $D_{10}$ x 71, $C_2\times C_{10}$ x 29, $C_{20}$ x 6, $C_5:C_4$ x 6

Order 16: $C_2\times D_4$ x 284, $C_2^4$ x 116, $D_4:C_2$ x 64, $C_2^2:C_4$ x 62, $C_2^2\times C_4$ x 48, $C_2\times Q_8$ x 6

Order 15: $C_{15}$

Order 14: $C_{14}$ x 3

Order 12: $D_6$ x 28, $C_2\times C_6$ x 7, $A_4$ x 5

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 8: $C_2^3$ x 216, $D_4$ x 112, $C_2\times C_4$ x 68, $Q_8$ x 6

Order 7: $C_7$

Order 6: $S_3$ x 8, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 100, $C_4$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 26880: $C_2^3:\GL(3,2)\times D_{10}$

Order 13440: $C_2^3:\GL(3,2)\times C_{10}$, $D_5\times C_2^3:\GL(3,2)$

Order 6720: $C_5\times C_2^3:\GL(3,2)$

Order 5376: $C_2^5:\GL(3,2)$

Order 3840: $C_2^3:S_4\times D_{10}$, $C_2^3:S_4\times D_{10}$

Order 3360: $D_5\times F_8:C_6$, $D_{10}\times \PSL(2,7)$

Order 2688: $C_2^4:\GL(3,2)$ x 2

Order 1920: $D_5\times C_2^3:S_4$ x 3, $D_5\times C_2^3:S_4$ x 2, $C_2^2:S_4\times D_{10}$, $Q_8:A_4\times D_{10}$, $(C_2^3\times C_{10}):S_4$, $C_2^3:A_4\times D_{10}$, $C_2^5:D_{30}$, $(C_2^2\times D_{10}):S_4$, $C_2^3:S_4\times C_{10}$, $C_2^3:S_4\times C_{10}$

Order 1680: $F_8:C_{30}$, $D_5\times F_8:C_3$, $C_{10}\times \GL(3,2)$, $D_5\times \GL(3,2)$

Order 1344: $C_2^3:\GL(3,2)$

Order 1280: $C_2\wr C_2^2\times D_{10}$

Order 1120: $D_{10}\times F_8$

Order 960: $C_5\times C_2^3:S_4$ x 3, $D_5\times C_2^3:A_4$ x 2, $C_2^4:D_{30}$ x 2, $(C_2\times D_{10}):S_4$ x 2, $D_5\times C_2^2:S_4$ x 2, $C_2\times D_{10}\times S_4$ x 2, $C_5\times C_2^3:S_4$ x 2, $C_2^5:C_{30}$, $(C_2^2\times D_{10}):A_4$, $C_2^2:A_4\times D_{10}$, $C_2^4:D_{30}$, $C_2^2:S_4\times C_{10}$, $C_{10}\times Q_8:A_4$, $D_5\times Q_8:A_4$, $(C_2^2\times C_{10}):S_4$

Order 840: $F_8:C_{15}$, $C_5\times \PSL(2,7)$

Order 768: $C_2^5:S_4$, $C_2^5:S_4$

Order 672: $C_2\times F_8:C_6$, $C_2^2\times \GL(3,2)$

Order 640: $C_2\wr C_2^2\times D_5$ x 6, $C_2^5:D_{10}$ x 2, $C_2^5:D_{10}$ x 2, $C_2^3:C_4\times D_{10}$ x 2, $C_2^4:D_{20}$ x 2, $D_{20}:C_2^4$, $C_2\wr C_2^2\times C_{10}$, $(D_4\times C_{10}):D_4$

Order 560: $C_{10}\times F_8$, $D_5\times F_8$

Order 480: $D_{10}\times S_4$ x 14, $C_2^4:C_{30}$ x 2, $C_2\times A_4\times D_{10}$ x 2, $C_5\times C_2^2:S_4$ x 2, $C_2^3:D_{30}$ x 2, $C_2\times C_{10}\times S_4$ x 2, $(C_2\times D_{10}):A_4$, $C_2^4:C_{30}$, $D_5\times C_2^2:A_4$, $C_2^4:D_{15}$, $C_5\times Q_8:A_4$, $D_{10}\times \SL(2,3)$

Order 420: $C_{70}:C_6$

Order 384: $C_2^4:S_4$ x 6, $C_2^4:S_4$ x 4, $C_2^2\wr S_3$, $C_2^5:A_4$, $C_2^5:A_4$

Order 336: $F_8:C_6$ x 2, $C_2\times \GL(3,2)$ x 2

Order 320: $C_2^4:D_{10}$ x 10, $C_2^4:D_{10}$ x 8, $C_2^3:C_4\times D_5$ x 7, $C_2^3:D_{20}$ x 7, $C_2\wr C_2^2\times C_5$ x 6, $D_{20}:C_2^3$ x 4, $(C_2\times C_{20}):D_4$ x 3, $C_2^3.D_{20}$ x 3, $C_{20}:C_2^4$ x 3, $C_2^4:D_{10}$ x 3, $C_2^3:D_{20}$ x 3, $C_2^4.D_{10}$ x 3, $C_2^4:C_{20}$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4:D_{10}$ x 2, $C_2^5:C_{10}$ x 2, $C_2^4:D_{10}$ x 2, $C_2^4\times D_{10}$, $C_{10}.C_2^5$, $D_{20}:C_2^3$, $D_{20}:C_2^3$

Order 280: $C_5\times F_8$

Order 256: $C_2^5:D_4$

Order 240: $D_5\times S_4$ x 14, $C_{10}\times S_4$ x 11, $A_4\times D_{10}$ x 9, $C_{10}:S_4$ x 8, $C_2^3:C_{30}$ x 2, $C_2^4:C_{15}$, $D_6\times D_{10}$, $C_{10}\times \SL(2,3)$, $D_5\times \SL(2,3)$

Order 224: $C_2^2\times F_8$

Order 210: $C_7:C_{30}$, $C_{35}:C_6$

Order 192: $C_2^3:S_4$ x 7, $C_2^3:S_4$ x 4, $C_2^4:A_4$ x 4, $C_2^3:S_4$ x 3, $C_2^3\times S_4$ x 2, $C_2^4:A_4$ x 2, $C_2^2\wr C_3$

Order 168: $F_8:C_3$, $\PSL(2,7)$

Order 160: $D_4\times D_{10}$ x 26, $D_{10}:D_4$ x 12, $D_4:D_{10}$ x 11, $C_2^4:C_{10}$ x 10, $C_2^2:D_{20}$ x 10, $D_{10}.D_4$ x 10, $C_2^3\times D_{10}$ x 9, $C_2^3:C_{20}$ x 7, $C_2^3:D_{10}$ x 6, $D_4:D_{10}$ x 6, $D_4:D_{10}$ x 6, $C_2^2.D_{20}$ x 6, $C_2^4:D_5$ x 5, $C_2^3.D_{10}$ x 4, $C_{10}.C_2^4$ x 4, $C_2^2.D_{20}$ x 4, $C_{20}:C_2^3$ x 3, $C_2^2\times D_{20}$ x 3, $C_{20}:C_2^3$ x 3, $C_2^3:C_{20}$ x 3, $C_2^3.D_{10}$ x 3, $D_4:D_{10}$ x 2, $D_{20}:C_2^2$ x 2, $C_2^4\times C_{10}$, $C_{10}.C_2^4$, $Q_8:D_{10}$, $Q_8\times D_{10}$

Order 140: $C_7\times D_{10}$

Order 128: $C_2^4:D_4$ x 12, $C_2^4:D_4$ x 2, $C_2^5:C_4$ x 2, $D_4:C_2^4$

Order 120: $C_5\times S_4$ x 8, $C_{10}\times A_4$ x 7, $S_3\times D_{10}$ x 5, $D_5\times A_4$ x 5, $C_5:S_4$ x 4, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_5\times \SL(2,3)$

Order 112: $C_2\times F_8$ x 2

Order 105: $C_7:C_{15}$

Order 96: $C_2^2\times S_4$ x 18, $C_2^3:A_4$ x 3, $C_2^2:S_4$ x 3, $C_2^3:A_4$ x 2, $C_2^3\times A_4$ x 2, $Q_8:A_4$, $C_2^2\times \SL(2,3)$

Order 84: $C_{14}:C_6$

Order 80: $C_2^2\times D_{10}$ x 42, $D_4\times D_5$ x 35, $C_{10}:D_4$ x 27, $D_4\times C_{10}$ x 21, $C_4\times D_{10}$ x 14, $C_2\times D_{20}$ x 13, $C_2^2:C_{20}$ x 10, $D_{20}:C_2$ x 8, $D_{10}:C_4$ x 8, $C_2^3\times C_{10}$ x 7, $D_4:D_5$ x 7, $D_4:C_{10}$ x 6, $C_{10}.D_4$ x 6, $C_2^2\times C_{20}$ x 3, $C_2^2.D_{10}$ x 3, $D_{20}:C_2$ x 3, $Q_8\times D_5$ x 2, $Q_8\times C_{10}$, $C_{10}:Q_8$

Order 70: $C_{70}$, $C_7\times D_5$

Order 64: $C_2\wr C_2^2$ x 24, $C_2^3:D_4$ x 20, $C_2^4:C_4$ x 14, $D_4:C_2^3$ x 8, $D_4\times C_2^3$ x 3, $C_2^4:C_4$ x 3, $C_2^6$, $D_4:C_2^3$

Order 60: $S_3\times D_5$ x 4, $S_3\times C_{10}$ x 4, $C_3\times D_{10}$ x 4, $C_5\times A_4$ x 3, $D_{30}$ x 3, $C_2\times C_{30}$

Order 56: $F_8$

Order 48: $C_2\times S_4$ x 33, $C_2^2\times A_4$ x 11, $C_2\times \SL(2,3)$ x 2, $C_2^2\times D_6$, $C_2^2:A_4$

Order 42: $C_7:C_6$ x 2

Order 40: $C_2\times D_{10}$ x 60, $C_2^2\times C_{10}$ x 21, $C_5\times D_4$ x 15, $C_5:D_4$ x 14, $C_2\times C_{20}$ x 11, $C_4\times D_5$ x 10, $C_{10}:C_4$ x 8, $D_{20}$ x 8, $C_5:Q_8$ x 2, $C_5\times Q_8$ x 2

Order 35: $C_{35}$

Order 32: $C_2^2\times D_4$ x 38, $C_2^2\wr C_2$ x 37, $D_4:C_2^2$ x 21, $C_2^3:C_4$ x 20, $C_2^3:C_4$ x 17, $D_4:C_2^2$ x 12, $C_2^5$ x 10, $C_2^3\times C_4$ x 3, $C_2^2\times Q_8$

Order 30: $C_{30}$ x 3, $C_3\times D_5$ x 2, $C_5\times S_3$ x 2, $D_{15}$

Order 28: $C_2\times C_{14}$

Order 24: $C_2\times A_4$ x 12, $S_4$ x 12, $C_2\times D_6$ x 7, $\SL(2,3)$, $C_2^2\times C_6$

Order 21: $C_7:C_3$

Order 20: $D_{10}$ x 25, $C_2\times C_{10}$ x 17, $C_{20}$ x 4, $C_5:C_4$ x 2

Order 16: $C_2\times D_4$ x 96, $C_2^4$ x 49, $C_2^2:C_4$ x 24, $D_4:C_2$ x 24, $C_2^2\times C_4$ x 20, $C_2\times Q_8$ x 4

Order 15: $C_{15}$

Order 14: $C_{14}$ x 2

Order 12: $D_6$ x 11, $C_2\times C_6$ x 5, $A_4$ x 3

Order 10: $C_{10}$ x 5, $D_5$ x 3

Order 8: $C_2^3$ x 81, $D_4$ x 37, $C_2\times C_4$ x 29, $Q_8$ x 4

Order 7: $C_7$

Order 6: $C_6$ x 5, $S_3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 42, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 26880: $C_2^3:\GL(3,2)\times D_{10}$ ($C_1$)

Order 13440: $D_5\times C_2^3:\GL(3,2)$ ($C_2$) x 2, $C_2^3:\GL(3,2)\times C_{10}$ ($C_2$)

Order 6720: $C_5\times C_2^3:\GL(3,2)$ ($C_2^2$)

Order 2688: $C_2^4:\GL(3,2)$ ($D_5$)

Order 1344: $C_2^3:\GL(3,2)$ ($D_{10}$)

Order 160: $C_2^3\times D_{10}$ ($\PSL(2,7)$)

Order 80: $C_2^2\times D_{10}$ ($C_2\times \GL(3,2)$) x 2, $C_2^3\times C_{10}$ ($C_2\times \GL(3,2)$)

Order 40: $C_2^2\times C_{10}$ ($C_2^2\times \GL(3,2)$)

Order 20: $D_{10}$ ($C_2^3:\GL(3,2)$)

Order 16: $C_2^4$ ($D_5\times \GL(3,2)$)

Order 10: $D_5$ ($C_2^4:\GL(3,2)$) x 2, $C_{10}$ ($C_2^4:\GL(3,2)$)

Order 8: $C_2^3$ ($D_{10}\times \PSL(2,7)$)

Order 5: $C_5$ ($C_2^5:\GL(3,2)$)

Order 2: $C_2$ ($D_5\times C_2^3:\GL(3,2)$)

Order 1: $C_1$ ($C_2^3:\GL(3,2)\times D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 26880: $C_2^3:\GL(3,2)\times D_{10}$ ($C_1$)

Order 13440: $C_2^3:\GL(3,2)\times C_{10}$ ($C_2$), $D_5\times C_2^3:\GL(3,2)$ ($C_2$)

Order 6720: $C_5\times C_2^3:\GL(3,2)$ ($C_2^2$)

Order 2688: $C_2^4:\GL(3,2)$ ($D_5$)

Order 1344: $C_2^3:\GL(3,2)$ ($D_{10}$)

Order 160: $C_2^3\times D_{10}$ ($\PSL(2,7)$)

Order 80: $C_2^3\times C_{10}$ ($C_2\times \GL(3,2)$), $C_2^2\times D_{10}$ ($C_2\times \GL(3,2)$)

Order 40: $C_2^2\times C_{10}$ ($C_2^2\times \GL(3,2)$)

Order 20: $D_{10}$ ($C_2^3:\GL(3,2)$)

Order 16: $C_2^4$ ($D_5\times \GL(3,2)$)

Order 10: $C_{10}$ ($C_2^4:\GL(3,2)$), $D_5$ ($C_2^4:\GL(3,2)$)

Order 8: $C_2^3$ ($D_{10}\times \PSL(2,7)$)

Order 5: $C_5$ ($C_2^5:\GL(3,2)$)

Order 2: $C_2$ ($D_5\times C_2^3:\GL(3,2)$)

Order 1: $C_1$ ($C_2^3:\GL(3,2)\times D_{10}$)

Series

Derived series

$C_2^3:\GL(3,2)\times D_{10}$

$\rhd$

$C_2^3:\GL(3,2)\times D_{10}$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

$\rhd$

$C_2^3:\GL(3,2)$

$\rhd$

$C_2^3:\GL(3,2)$

Chief series

$C_2^3:\GL(3,2)\times D_{10}$

$\rhd$

$C_2^3:\GL(3,2)\times D_{10}$

$\rhd$

$C_2^3\times D_{10}$

$\rhd$

$C_2^3\times D_{10}$

$\rhd$

$C_2^2\times D_{10}$

$\rhd$

$D_{10}$

$\rhd$

$D_{10}$

$\rhd$

$C_2^2\times C_{10}$

$\rhd$

$D_5$

$\rhd$

$D_5$

$\rhd$

$C_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2^3:\GL(3,2)\times D_{10}$

$\rhd$

$C_2^3:\GL(3,2)\times D_{10}$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 2 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 $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.