Abstract group 43008.bx: $C_2^8.\GL(3,2)$

• Label: 43008.bx
• Order: \( 2^{11} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$C_2^8.\GL(3,2)$
Order:	\(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^8.S_3.\PSL(2,7)\times S_3$, of order \(1548288\)\(\medspace = 2^{13} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 8, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	14
Elements	1	1599	896	14784	13440	3072	9216	43008
Conjugacy classes	1	35	1	40	15	2	6	100
Divisions	1	35	1	40	15	1	3	96
Autjugacy classes	1	7	1	4	3	2	2	20

Dimension	1	3	6	7	8	14	21	42
Irr. complex chars.	4	8	4	28	4	12	24	16	100
Irr. rational chars.	4	0	8	28	4	12	24	16	96

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $20$ $\langle(1,2)(3,4)(5,7)(6,8)(9,10)(11,12,13,14)(15,17,16,18)(19,20), (1,3,5,4,6,2)(7,8)(11,13,15,17,18,12)(14,16)(19,20)\rangle$
Transitive group:	28T536				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_2^6:\GL(3,2))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^8$ . $\PSL(2,7)$	$(C_2^7:\GL(3,2))$ . $C_2$	$C_2^7$ . $(C_2\times \GL(3,2))$	$C_2$ . $(C_2^7:\GL(3,2))$	all 10
Aut. group:	$\Aut((C_4\times D_4):D_6)$

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 11718715 subgroups in 145428 conjugacy classes, 30 normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^6:\GL(3,2)$
Commutator:	$G' \simeq$ $C_2^6:\GL(3,2)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^8.\GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8$	$G/\operatorname{Fit} \simeq$ $\PSL(2,7)$
Radical:	$R \simeq$ $C_2^8$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_2^8$	$G/\operatorname{soc} \simeq$ $\PSL(2,7)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 384 (order at least 112) are shown, as well as all normal subgroups of any index.

Order 43008: $C_2^8.\GL(3,2)$

Order 21504: $C_2^7:\GL(3,2)$ x 3

Order 10752: $C_2^6:\GL(3,2)$

Order 6144: $C_2^4.C_2^4.D_6.C_2$, $C_2^8.S_4$

Order 5376: $C_2^5:\GL(3,2)$ x 6, $C_2^5:F_8:C_3$

Order 3072: $C_2^7:S_4$ x 24, $C_2^4.C_2^4.D_6$ x 6, $C_2^7:S_4$ x 6, $C_2^4.C_2^4.C_6.C_2$, $C_2^8.A_4$

Order 2688: $C_2^4:\GL(3,2)$ x 18, $C_2^3:F_8:C_6$ x 3

Order 2048: $C_2^8.D_4$

Order 1792: $C_2^5:F_8$

Order 1536: $C_2^6:S_4$ x 72, $C_2^6:S_4$ x 64, $C_2^7:A_4$ x 12, $C_2^6:S_4$ x 6, $C_2^6:S_4$ x 4, $C_2^6:S_4$ x 4, $C_2^7.A_4$ x 3, $C_2^7:A_4$ x 3, $C_2^6:S_4$

Order 1344: $C_2^3:\GL(3,2)$ x 6, $C_2^3:F_8:C_3$

Order 1024: $C_2^7:D_4$ x 48, $C_2^6.C_2^4$ x 6, $C_2^7:D_4$ x 6, $C_2^2\times C_2^3.C_2^5$, $C_2^7.C_2^3$, $C_2^6.C_2^4$

Order 896: $C_2^4:F_8$ x 3

Order 768: $C_2^5:S_4$ x 96, $C_2^5:S_4$ x 84, $C_2^5:S_4$ x 30, $C_2^5:S_4$ x 24, $C_2^6:A_4$ x 18, $C_2^6:A_4$ x 16, $C_2^5:S_4$ x 6, $C_2^6:A_4$ x 6, $C_2^6.A_4$ x 4, $C_2^2\times (C_2\times Q_8):A_4$ x 3, $C_2^8.C_3$, $C_2^6:A_4$, $C_2^6:A_4$

Order 672: $C_2^2\times \GL(3,2)$ x 4, $C_2\times F_8:C_6$ x 3

Order 512: $C_2^6:D_4$ x 256, $C_2^6:D_4$ x 216, $C_2^5.C_2^4$ x 144, $C_4^2:C_2^5$ x 45, $C_2^5.C_2^4$ x 24, $C_2^4.C_2^5$ x 24, $C_2^4.C_2^5$ x 12, $C_2^2\times C_2^5.C_2^2$ x 12, $C_2^4.C_2^5$ x 6, $C_2^4.C_2^5$ x 6, $C_4^2:C_2^5$ x 6, $C_2^6:D_4$ x 6, $C_2^8.C_2$ x 4, $D_4^2.C_2^3$ x 4, $C_2^6:D_4$ x 4, $C_2^4.C_2^5$ x 3, $C_2^2\times C_2^4.C_2^3$ x 3

Order 448: $C_2^3:F_8$

Order 384: $C_2^4:S_4$ x 168, $C_2^2\wr S_3$ x 152, $C_2^4:S_4$ x 72, $C_2^5:A_4$ x 42, $C_2^4:S_4$ x 16, $C_2^5:A_4$ x 15, $C_2^5:A_4$ x 12, $C_2^5:A_4$ x 12, $C_2^4:\SL(2,3)$ x 9, $C_2^4\times S_4$ x 7, $C_2^5:A_4$ x 6, $C_2^5:A_4$ x 3

Order 336: $C_2\times \GL(3,2)$ x 12, $F_8:C_6$ x 9

Order 256: $C_2^5:D_4$ x 960, $C_4^2:C_2^4$ x 715, $C_4^2:C_2^4$ x 384, $C_2^5.D_4$ x 384, $C_4^2:C_2^4$ x 360, $C_4^2:C_2^4$ x 288, $C_2^5:D_4$ x 288, $C_2^5:D_4$ x 240, $C_2^5.D_4$ x 216, $C_2^5:D_4$ x 172, $C_2^5.D_4$ x 144, $C_2^5:D_4$ x 120, $C_2^6:C_4$ x 108, $C_2^6:C_2^2$ x 80, $C_2^6:C_4$ x 64, $C_2^5.D_4$ x 54, $C_4^2:C_2^4$ x 24, $C_2^6:C_4$ x 24, $C_2^6:C_4$ x 16, $C_2^5:D_4$ x 15, $C_4^2:C_2^4$ x 12, $C_2^2\times D_4^2$ x 12, $C_4^2:C_2^4$ x 12, $C_2^4.C_4^2$ x 12, $D_4:C_2^5$ x 6, $C_4^2:C_2^4$ x 6, $C_4^2:C_2^4$ x 6, $C_4^2:C_2^4$ x 6, $C_2^6:C_4$ x 6, $C_4^2:C_2^4$ x 4, $C_4^2:C_2^4$ x 4, $D_4\times C_2^5$ x 3, $C_2^5:Q_8$ x 3, $C_2^8$ x 2, $C_2^4.C_4^2$

Order 224: $C_2^2\times F_8$ x 5

Order 192: $C_2^3\times S_4$ x 210, $C_2^3:S_4$ x 192, $C_2^3:S_4$ x 96, $C_2^3:S_4$ x 48, $C_2^2\wr C_3$ x 42, $C_2^4:A_4$ x 42, $C_2^4:A_4$ x 36, $A_4\times C_2^4$ x 8, $C_2^3\times \SL(2,3)$ x 6, $C_2^4:A_4$ x 4, $C_2^3:\SL(2,3)$ x 3, $C_2^4:A_4$

Order 168: $\PSL(2,7)$ x 4, $F_8:C_3$ x 3

Order 128: $D_4^2:C_2$ x 4096, $D_4^2:C_2$ x 3264, $C_4^2:C_2^3$ x 3072, $C_2^4:D_4$ x 2470, $C_2^4:D_4$ x 1536, $C_4^2:C_2^3$ x 1152, $C_2^4:D_4$ x 768, $C_4^2:C_2^3$ x 576, $C_2\times D_4^2$ x 576, $C_4^2:C_2^3$ x 576, $C_2^4:D_4$ x 504, $C_2^4.D_4$ x 480, $C_2^5:C_4$ x 456, $C_2^3\wr C_2$ x 448, $C_2^4.D_4$ x 384, $C_4^2:C_2^3$ x 288, $C_4^2:C_2^3$ x 288, $C_2^5:C_4$ x 276, $C_2^5:C_4$ x 240, $C_2^5:C_4$ x 219, $C_4^2:C_2^3$ x 192, $D_4\times C_2^4$ x 184, $C_4^2:C_2^3$ x 180, $C_2^3.C_4^2$ x 144, $D_4:C_2^4$ x 138, $C_2^4.D_4$ x 120, $C_4^2:C_2^3$ x 72, $C_2^7$ x 54, $C_2^4:Q_8$ x 54, $C_2^4:D_4$ x 36, $C_2^4.D_4$ x 24, $C_4^2:C_2^3$ x 24, $C_2^3.C_4^2$ x 16, $C_4^2:C_2^3$ x 12, $C_2^4:Q_8$ x 12, $C_4^2:C_2^3$ x 8, $D_4:C_2^4$ x 6, $C_4^2:C_2^3$ x 6, $C_4^2:C_2^3$ x 4, $C_2^5\times C_4$ x 3

Order 112: $C_2\times F_8$ x 15

Order 64: $C_2^6$

Order 32: $C_2^5$ x 3

Order 16: $C_2^4$ x 9

Order 8: $C_2^3$ x 3

Order 7: $C_7$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 384 (order at least 112) are shown, as well as all normal subgroups of any index.

Order 43008: $C_2^8.\GL(3,2)$

Order 21504: $C_2^7:\GL(3,2)$

Order 10752: $C_2^6:\GL(3,2)$

Order 6144: $C_2^4.C_2^4.D_6.C_2$, $C_2^8.S_4$

Order 5376: $C_2^5:\GL(3,2)$, $C_2^5:F_8:C_3$

Order 3072: $C_2^7:S_4$ x 3, $C_2^4.C_2^4.D_6$ x 2, $C_2^4.C_2^4.C_6.C_2$, $C_2^8.A_4$, $C_2^7:S_4$

Order 2688: $C_2^3:F_8:C_6$, $C_2^4:\GL(3,2)$

Order 2048: $C_2^8.D_4$

Order 1792: $C_2^5:F_8$

Order 1536: $C_2^6:S_4$ x 5, $C_2^6:S_4$ x 3, $C_2^6:S_4$ x 2, $C_2^7:A_4$ x 2, $C_2^6:S_4$, $C_2^6:S_4$, $C_2^6:S_4$, $C_2^7.A_4$, $C_2^7:A_4$

Order 1344: $C_2^3:F_8:C_3$, $C_2^3:\GL(3,2)$

Order 1024: $C_2^7:D_4$ x 6, $C_2^2\times C_2^3.C_2^5$, $C_2^7.C_2^3$, $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^7:D_4$

Order 896: $C_2^4:F_8$

Order 768: $C_2^5:S_4$ x 6, $C_2^5:S_4$ x 4, $C_2^5:S_4$ x 4, $C_2^6:A_4$ x 3, $C_2^5:S_4$ x 2, $C_2^6:A_4$ x 2, $C_2^5:S_4$, $C_2^8.C_3$, $C_2^6:A_4$, $C_2^6:A_4$, $C_2^6:A_4$, $C_2^6.A_4$, $C_2^2\times (C_2\times Q_8):A_4$

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

Order 512: $C_2^6:D_4$ x 13, $C_2^6:D_4$ x 9, $C_2^5.C_2^4$ x 5, $C_4^2:C_2^5$ x 5, $C_2^4.C_2^5$ x 4, $C_2^5.C_2^4$ x 3, $C_2^6:D_4$ x 2, $C_2^8.C_2$, $C_2^4.C_2^5$, $C_2^2\times C_2^5.C_2^2$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $D_4^2.C_2^3$, $C_2^2\times C_2^4.C_2^3$, $C_4^2:C_2^5$, $C_2^6:D_4$

Order 448: $C_2^3:F_8$

Order 384: $C_2^2\wr S_3$ x 10, $C_2^4:S_4$ x 9, $C_2^5:A_4$ x 3, $C_2^5:A_4$ x 3, $C_2^4:S_4$ x 3, $C_2^4:S_4$ x 2, $C_2^4\times S_4$ x 2, $C_2^5:A_4$ x 2, $C_2^5:A_4$, $C_2^5:A_4$, $C_2^4:\SL(2,3)$, $C_2^5:A_4$

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

Order 256: $C_2^5:D_4$ x 25, $C_4^2:C_2^4$ x 24, $C_2^5:D_4$ x 18, $C_2^5:D_4$ x 14, $C_4^2:C_2^4$ x 12, $C_2^6:C_2^2$ x 10, $C_4^2:C_2^4$ x 10, $C_4^2:C_2^4$ x 10, $C_2^5.D_4$ x 9, $C_2^5:D_4$ x 8, $C_2^5:D_4$ x 6, $C_2^5.D_4$ x 6, $C_2^5.D_4$ x 6, $C_2^5.D_4$ x 5, $C_2^6:C_4$ x 5, $C_2^6:C_4$ x 5, $C_2^6:C_4$ x 3, $C_2^8$ x 2, $C_2^5:D_4$ x 2, $C_2^6:C_4$ x 2, $C_2^6:C_4$ x 2, $D_4:C_2^5$, $D_4\times C_2^5$, $C_4^2:C_2^4$, $C_4^2:C_2^4$, $C_2^2\times D_4^2$, $C_4^2:C_2^4$, $C_4^2:C_2^4$, $C_4^2:C_2^4$, $C_4^2:C_2^4$, $C_2^5:Q_8$, $C_4^2:C_2^4$, $C_4^2:C_2^4$, $C_2^4.C_4^2$, $C_2^4.C_4^2$

Order 224: $C_2^2\times F_8$ x 2

Order 192: $C_2^3\times S_4$ x 12, $C_2^2\wr C_3$ x 8, $C_2^3:S_4$ x 8, $C_2^3:S_4$ x 4, $C_2^3:S_4$ x 4, $A_4\times C_2^4$ x 3, $C_2^4:A_4$ x 3, $C_2^4:A_4$ x 2, $C_2^4:A_4$, $C_2^3\times \SL(2,3)$, $C_2^4:A_4$, $C_2^3:\SL(2,3)$

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

Order 128: $C_2^4:D_4$ x 111, $D_4^2:C_2$ x 51, $C_2^4:D_4$ x 51, $D_4^2:C_2$ x 47, $C_4^2:C_2^3$ x 42, $C_2^5:C_4$ x 20, $C_2^5:C_4$ x 19, $D_4\times C_2^4$ x 18, $C_4^2:C_2^3$ x 16, $C_2^4:D_4$ x 15, $C_2^4:D_4$ x 15, $C_2^3\wr C_2$ x 13, $C_2^7$ x 12, $C_2^4.D_4$ x 11, $C_2^4.D_4$ x 11, $C_4^2:C_2^3$ x 10, $C_2\times D_4^2$ x 10, $C_4^2:C_2^3$ x 10, $C_2^5:C_4$ x 9, $C_4^2:C_2^3$ x 9, $C_4^2:C_2^3$ x 9, $C_2^4.D_4$ x 9, $D_4:C_2^4$ x 8, $C_2^5:C_4$ x 8, $C_4^2:C_2^3$ x 6, $C_4^2:C_2^3$ x 6, $C_2^3.C_4^2$ x 4, $C_2^3.C_4^2$ x 4, $C_4^2:C_2^3$ x 4, $C_2^4:Q_8$ x 3, $C_2^4.D_4$ x 2, $C_2^4:D_4$ x 2, $D_4:C_2^4$, $C_2^5\times C_4$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_2^4:Q_8$, $C_4^2:C_2^3$, $C_4^2:C_2^3$

Order 112: $C_2\times F_8$ x 2

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 7: $C_7$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 43008: $C_2^8.\GL(3,2)$ ($C_1$)

Order 21504: $C_2^7:\GL(3,2)$ ($C_2$) x 3

Order 10752: $C_2^6:\GL(3,2)$ ($C_2^2$)

Order 256: $C_2^8$ ($\PSL(2,7)$)

Order 128: $C_2^7$ ($C_2\times \GL(3,2)$) x 3

Order 64: $C_2^6$ ($C_2^2\times \GL(3,2)$)

Order 32: $C_2^5$ ($C_2^3:\GL(3,2)$) x 3

Order 16: $C_2^4$ ($C_2^4:\GL(3,2)$) x 9

Order 8: $C_2^3$ ($C_2^5:\GL(3,2)$) x 3

Order 4: $C_2^2$ ($C_2^6:\GL(3,2)$)

Order 2: $C_2$ ($C_2^7:\GL(3,2)$) x 3

Order 1: $C_1$ ($C_2^8.\GL(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 43008: $C_2^8.\GL(3,2)$ ($C_1$)

Order 21504: $C_2^7:\GL(3,2)$ ($C_2$)

Order 10752: $C_2^6:\GL(3,2)$ ($C_2^2$)

Order 256: $C_2^8$ ($\PSL(2,7)$)

Order 128: $C_2^7$ ($C_2\times \GL(3,2)$)

Order 64: $C_2^6$ ($C_2^2\times \GL(3,2)$)

Order 32: $C_2^5$ ($C_2^3:\GL(3,2)$)

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

Order 8: $C_2^3$ ($C_2^5:\GL(3,2)$)

Order 4: $C_2^2$ ($C_2^6:\GL(3,2)$)

Order 2: $C_2$ ($C_2^7:\GL(3,2)$)

Order 1: $C_1$ ($C_2^8.\GL(3,2)$)

Series

Derived series

$C_2^8.\GL(3,2)$

$\rhd$

$C_2^6:\GL(3,2)$

Chief series

$C_2^8.\GL(3,2)$

$\rhd$

$C_2^8$

$\rhd$

$C_2^5$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^8.\GL(3,2)$

$\rhd$

$C_2^6:\GL(3,2)$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 3 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 $100 \times 100$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $96 \times 96$ rational character table.