Abstract group 17635968.c: $C_3^8:C_2^4.\GL(3,2)$

• Label: 17635968.c
• Order: \( 2^{7} \cdot 3^{9} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{9} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_3^8:C_2^4.\GL(3,2)$
Order:	\(17635968\)\(\medspace = 2^{7} \cdot 3^{9} \cdot 7 \)
Exponent:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Automorphism group:	$C_3^8.C_2^5.\PSL(2,7)$, of order \(35271936\)\(\medspace = 2^{8} \cdot 3^{9} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 8, $\PSL(2,7)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	9	12	14	18	21
Elements	1	45495	169856	503496	3604608	279936	1306368	5007744	2519424	1959552	2239488	17635968
Conjugacy classes	1	7	37	6	108	2	21	42	2	6	8	240
Divisions	1	7	37	6	108	1	21	42	1	6	4	234
Autjugacy classes	1	7	31	6	90	2	16	34	2	5	6	200

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $24$ $\langle(1,16,5,9,11,23,2,17,4,8,12,22,3,18,6,7,10,24)(13,21)(14,20)(15,19), (1,10,9,4,21,23,2,11,7,6,19,24,3,12,8,5,20,22)(13,17,14,16,15,18)\rangle$
Transitive group:	24T24098	42T2865			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^8:C_2^4)$ . $\PSL(2,7)$	$C_3^8$ . $(C_2^4.\PSL(2,7))$	$(C_3^7.C_2^3:\GL(3,2))$ . $S_3$	$C_3^7$ . $(S_3\times C_2^3:\GL(3,2))$	all 9

Elements of the group are displayed as permutations of degree 24.

Homology

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

Subgroups

There are 11 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^8:C_2^4.\GL(3,2)$
Commutator:	$G' \simeq$ $C_3^8.C_2^3:\GL(3,2)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^8:C_2^4.\GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^8$	$G/\operatorname{Fit} \simeq$ $C_2^4.\PSL(2,7)$
Radical:	$R \simeq$ $C_3^8:C_2^4$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_3^8$	$G/\operatorname{soc} \simeq$ $C_2^4.\PSL(2,7)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8.C_3$
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 112 (order at least 157464) are shown, as well as all normal subgroups of any index.

Order 17635968: $C_3^8:C_2^4.\GL(3,2)$

Order 8817984: $C_3^8.C_2^3:\GL(3,2)$

Order 5878656: $C_3^7.C_2^4:\GL(3,2)$

Order 2939328: $C_3^7.C_2^3:\GL(3,2)$

Order 2519424: $C_3^6.D_6^2:S_4$, $C_3^8.C_2^4:S_4$

Order 2204496: $C_3^8.C_2.\PSL(2,7)$, $C_3^7.(S_3\times \GL(3,2))$, $C_3^8.F_8:C_6$

Order 1259712: $C_3^8.C_2^3:S_4$ x 2, $C_3^8.C_2^3:S_4$ x 2, $C_3^6.C_2^4.C_3^2.D_6$, $C_3^5.C_6^3:S_4$, $C_3^5.C_6^3:S_4$, $C_3^8.C_2^3:S_4$, $C_3^6.D_6^2:A_4$, $C_3^8.C_2^4:A_4$

Order 1102248: $C_3^8.\PSL(2,7)$, $C_3^8.\PSL(2,7)$, $C_3^8.F_8:C_3$

Order 839808: $C_3^7.C_2^4:S_4$ x 2, $C_3^7.C_2^4:S_4$, $C_3^3.S_3^5:C_2^2$

Order 734832: $C_3^6.(S_3\times \GL(3,2))$ x 4, $C_3^8:(C_2\times F_8)$, $C_3^7.(C_2\times \GL(3,2))$, $C_3^7.F_8:C_6$

Order 629856: $C_3^6.(S_4\times S_3^2)$ x 2, $C_3^2\wr C_2^2.S_4$ x 2, $C_3^8.C_2^3:A_4$ x 2, $C_3^5.A_4.C_3^3.C_2^3$, $C_3^6.C_2^4.C_3^3.C_2$, $C_3^7.C_2.C_2^4.C_3^2$, $C_3^8.C_2^4.C_6$, $C_3^6.C_2^4.C_3^3.C_2$

Order 419904: $C_3^7.C_2^3:S_4$ x 4, $C_3^4.S_3^3:S_4$ x 3, $C_3^8.C_2^5.C_2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^4:A_4$ x 2, $C_3^6.D_6^2:C_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^8.C_2^4.C_2^2$, $C_3^7.C_2^4:A_4$, $C_3^2\wr C_2^2.C_2^4$, $C_3^8:(C_2^4:C_4)$, $C_3^4.C_6^3:S_4$

Order 367416: $C_3^7.\PSL(2,7)$ x 4, $C_3^7.F_8:C_3$ x 2, $C_3^8:F_8$, $C_3^7.\GL(3,2)$

Order 314928: $C_3^7.(S_3\times S_4)$ x 4, $C_3^7.(C_6\times S_4)$ x 4, $C_3^7.(S_3\times S_4)$ x 3, $C_3^7.(S_3\times S_4)$ x 2, $C_3^7.(A_4\times D_6)$ x 2, $C_3^5.A_4.C_3^2.D_6$, $C_3^4.C_6^2.C_3^2.D_6$, $C_3^5.C_6^2.S_3^2$, $C_3^5.C_6^2.C_3.D_6$, $C_3^5.A_4.C_3^3.C_2^2$, $C_3^4.C_6^2.C_3^2.D_6$, $C_3^4.C_6^2.C_3.C_6^2$, $C_3^6.C_2^4.C_3^3$, $C_3^8.A_4.C_2^2$, $C_3^7.(S_3\times \SL(2,3))$

Order 279936: $C_3^2.S_3^5:C_2^2$ x 2, $C_3^6.C_2^4:S_4$

Order 275562: $C_3^6.C_{21}.C_3.C_6$

Order 244944: $C_3^7.F_8.C_2$, $C_3^6:(C_2\times \GL(3,2))$

Order 209952: $C_3^2\wr C_2^2.D_4$ x 6, $C_3^7.C_2^2:S_4$ x 6, $C_3^7.C_2^3:A_4$ x 4, $C_3^2\wr C_2^2.D_4$ x 4, $C_3^2\wr C_2^2.D_4$ x 4, $C_3^2\wr C_2^2.D_4$ x 4, $C_3^6.(D_6\times S_4)$ x 4, $C_3^7.C_2^2:S_4$ x 4, $C_3^2\wr C_2^2.C_2^3$ x 3, $C_3^7.C_2^2:S_4$ x 3, $C_3^4.S_3^3:A_4$ x 3, $C_3\wr S_4.D_6$ x 3, $C_3^2\wr C_2^2.D_4$ x 3, $C_3^8:(C_2^2\times D_4)$ x 3, $C_3^7.C_2^2:S_4$ x 2, $C_3^7.C_2^3:A_4$ x 2, $C_3^4.C_6^3:A_4$ x 2, $C_3^2\wr C_2^2.C_2^3$ x 2, $C_3^2\wr C_2^2.D_4$ x 2, $C_3^2\wr C_2^2.D_4$ x 2, $C_3^8.C_2^4.C_2$, $C_3^4.C_6^2.S_3^2.C_2$, $C_3^8.C_2^4.C_2$, $C_3^8.C_2^5$, $C_3^5.C_6^3.C_2^2$, $C_3^8.C_2^2\wr C_2$, $C_3^5.C_6^3.C_2^2$, $C_3^7.C_2^3:D_6$, $C_3^5.C_6^2.D_6.C_2$, $C_3^8.C_2^2\wr C_2$, $C_3^7.C_2^4:S_3$, $C_3^8.C_2^3.C_2^2$, $C_3^8.C_2^3.C_2^2$, $C_3^2\wr C_2^2.C_2^3$

Order 157464: $C_3^8.S_4$ x 6, $C_3^7.(C_6\times A_4)$ x 4, $C_3^7.(S_3\times A_4)$ x 3, $C_3^5.C_6^2.C_3.S_3$ x 2, $C_3^3.C_6^2.C_3^4.C_2$ x 2, $C_3^8.A_4.C_2$ x 2, $C_3^8.S_4$ x 2, $C_3^8.S_4$ x 2, $C_3^3.C_6^2.\He_3.C_6$, $C_3^6.S_3^3$, $C_3^6.C_6^2.C_6$, $C_3^6.C_6^2.C_6$, $C_3^5.C_6^2.C_3.C_6$, $C_3^8.\SL(2,3)$

Order 104976: $C_3^8:C_2^4$

Order 52488: $C_3^8:C_2^3$

Order 19683: $C_3^8.C_3$

Order 17496: $C_3^7.C_2^3$

Order 13122: $C_3^8.C_2$

Order 8064: $(C_2^3\times S_3).\PSL(2,7)$

Order 6561: $C_3^8$

Order 2187: $C_3^7$

Order 128: $C_2^4:D_4$

Order 7: $C_7$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 112 (order at least 157464) are shown, as well as all normal subgroups of any index.

Order 17635968: $C_3^8:C_2^4.\GL(3,2)$

Order 8817984: $C_3^8.C_2^3:\GL(3,2)$

Order 5878656: $C_3^7.C_2^4:\GL(3,2)$

Order 2939328: $C_3^7.C_2^3:\GL(3,2)$

Order 2519424: $C_3^6.D_6^2:S_4$, $C_3^8.C_2^4:S_4$

Order 2204496: $C_3^8.C_2.\PSL(2,7)$, $C_3^7.(S_3\times \GL(3,2))$, $C_3^8.F_8:C_6$

Order 1259712: $C_3^8.C_2^3:S_4$ x 2, $C_3^8.C_2^3:S_4$ x 2, $C_3^6.C_2^4.C_3^2.D_6$, $C_3^5.C_6^3:S_4$, $C_3^5.C_6^3:S_4$, $C_3^8.C_2^3:S_4$, $C_3^6.D_6^2:A_4$, $C_3^8.C_2^4:A_4$

Order 1102248: $C_3^8.\PSL(2,7)$, $C_3^8.\PSL(2,7)$, $C_3^8.F_8:C_3$

Order 839808: $C_3^7.C_2^4:S_4$ x 2, $C_3^7.C_2^4:S_4$, $C_3^3.S_3^5:C_2^2$

Order 734832: $C_3^6.(S_3\times \GL(3,2))$ x 3, $C_3^8:(C_2\times F_8)$, $C_3^7.(C_2\times \GL(3,2))$, $C_3^7.F_8:C_6$

Order 629856: $C_3^6.(S_4\times S_3^2)$ x 2, $C_3^2\wr C_2^2.S_4$ x 2, $C_3^8.C_2^3:A_4$ x 2, $C_3^5.A_4.C_3^3.C_2^3$, $C_3^6.C_2^4.C_3^3.C_2$, $C_3^7.C_2.C_2^4.C_3^2$, $C_3^8.C_2^4.C_6$, $C_3^6.C_2^4.C_3^3.C_2$

Order 419904: $C_3^7.C_2^3:S_4$ x 4, $C_3^4.S_3^3:S_4$ x 3, $C_3^8.C_2^5.C_2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^8:C_2\wr C_2^2$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^4:A_4$ x 2, $C_3^6.D_6^2:C_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^7.C_2^3:S_4$ x 2, $C_3^8.C_2^4.C_2^2$, $C_3^7.C_2^4:A_4$, $C_3^2\wr C_2^2.C_2^4$, $C_3^8:(C_2^4:C_4)$, $C_3^4.C_6^3:S_4$

Order 367416: $C_3^7.\PSL(2,7)$ x 3, $C_3^7.F_8:C_3$ x 2, $C_3^8:F_8$, $C_3^7.\GL(3,2)$

Order 314928: $C_3^7.(S_3\times S_4)$ x 4, $C_3^7.(C_6\times S_4)$ x 4, $C_3^7.(S_3\times S_4)$ x 3, $C_3^7.(S_3\times S_4)$ x 2, $C_3^7.(A_4\times D_6)$ x 2, $C_3^5.A_4.C_3^2.D_6$, $C_3^4.C_6^2.C_3^2.D_6$, $C_3^5.C_6^2.S_3^2$, $C_3^5.C_6^2.C_3.D_6$, $C_3^5.A_4.C_3^3.C_2^2$, $C_3^4.C_6^2.C_3^2.D_6$, $C_3^4.C_6^2.C_3.C_6^2$, $C_3^6.C_2^4.C_3^3$, $C_3^8.A_4.C_2^2$, $C_3^7.(S_3\times \SL(2,3))$

Order 279936: $C_3^2.S_3^5:C_2^2$ x 2, $C_3^6.C_2^4:S_4$

Order 275562: $C_3^6.C_{21}.C_3.C_6$

Order 244944: $C_3^7.F_8.C_2$, $C_3^6:(C_2\times \GL(3,2))$

Order 209952: $C_3^2\wr C_2^2.D_4$ x 6, $C_3^7.C_2^2:S_4$ x 6, $C_3^7.C_2^3:A_4$ x 4, $C_3^2\wr C_2^2.D_4$ x 4, $C_3^2\wr C_2^2.D_4$ x 4, $C_3^2\wr C_2^2.D_4$ x 4, $C_3^6.(D_6\times S_4)$ x 4, $C_3^7.C_2^2:S_4$ x 4, $C_3^2\wr C_2^2.C_2^3$ x 3, $C_3^7.C_2^2:S_4$ x 3, $C_3^4.S_3^3:A_4$ x 3, $C_3^2\wr C_2^2.D_4$ x 3, $C_3^8:(C_2^2\times D_4)$ x 3, $C_3^7.C_2^2:S_4$ x 2, $C_3^7.C_2^3:A_4$ x 2, $C_3^4.C_6^3:A_4$ x 2, $C_3^2\wr C_2^2.C_2^3$ x 2, $C_3^2\wr C_2^2.D_4$ x 2, $C_3^2\wr C_2^2.D_4$ x 2, $C_3\wr S_4.D_6$ x 2, $C_3^8.C_2^4.C_2$, $C_3^4.C_6^2.S_3^2.C_2$, $C_3^8.C_2^4.C_2$, $C_3^8.C_2^5$, $C_3^5.C_6^3.C_2^2$, $C_3^8.C_2^2\wr C_2$, $C_3^5.C_6^3.C_2^2$, $C_3^7.C_2^3:D_6$, $C_3^5.C_6^2.D_6.C_2$, $C_3^8.C_2^2\wr C_2$, $C_3^7.C_2^4:S_3$, $C_3^8.C_2^3.C_2^2$, $C_3^8.C_2^3.C_2^2$, $C_3^2\wr C_2^2.C_2^3$

Order 157464: $C_3^8.S_4$ x 6, $C_3^7.(C_6\times A_4)$ x 4, $C_3^7.(S_3\times A_4)$ x 3, $C_3^5.C_6^2.C_3.S_3$ x 2, $C_3^3.C_6^2.C_3^4.C_2$ x 2, $C_3^8.A_4.C_2$ x 2, $C_3^8.S_4$ x 2, $C_3^8.S_4$ x 2, $C_3^3.C_6^2.\He_3.C_6$, $C_3^6.S_3^3$, $C_3^6.C_6^2.C_6$, $C_3^6.C_6^2.C_6$, $C_3^5.C_6^2.C_3.C_6$, $C_3^8.\SL(2,3)$

Order 104976: $C_3^8:C_2^4$

Order 52488: $C_3^8:C_2^3$

Order 19683: $C_3^8.C_3$

Order 17496: $C_3^7.C_2^3$

Order 13122: $C_3^8.C_2$

Order 8064: $(C_2^3\times S_3).\PSL(2,7)$

Order 6561: $C_3^8$

Order 2187: $C_3^7$

Order 128: $C_2^4:D_4$

Order 7: $C_7$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 17635968: $C_3^8:C_2^4.\GL(3,2)$ ($C_1$)

Order 8817984: $C_3^8.C_2^3:\GL(3,2)$ ($C_2$)

Order 2939328: $C_3^7.C_2^3:\GL(3,2)$ ($S_3$)

Order 104976: $C_3^8:C_2^4$ ($\PSL(2,7)$)

Order 52488: $C_3^8:C_2^3$ ($C_2\times \GL(3,2)$)

Order 17496: $C_3^7.C_2^3$ ($S_3\times \GL(3,2)$)

Order 13122: $C_3^8.C_2$ ($C_2^3:\GL(3,2)$)

Order 6561: $C_3^8$ ($C_2^4.\PSL(2,7)$)

Order 2187: $C_3^7$ ($S_3\times C_2^3:\GL(3,2)$)

Order 3: $C_3$ ($C_3^7.C_2^4:\GL(3,2)$)

Order 1: $C_1$ ($C_3^8:C_2^4.\GL(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 17635968: $C_3^8:C_2^4.\GL(3,2)$ ($C_1$)

Order 8817984: $C_3^8.C_2^3:\GL(3,2)$ ($C_2$)

Order 2939328: $C_3^7.C_2^3:\GL(3,2)$ ($S_3$)

Order 104976: $C_3^8:C_2^4$ ($\PSL(2,7)$)

Order 52488: $C_3^8:C_2^3$ ($C_2\times \GL(3,2)$)

Order 17496: $C_3^7.C_2^3$ ($S_3\times \GL(3,2)$)

Order 13122: $C_3^8.C_2$ ($C_2^3:\GL(3,2)$)

Order 6561: $C_3^8$ ($C_2^4.\PSL(2,7)$)

Order 2187: $C_3^7$ ($S_3\times C_2^3:\GL(3,2)$)

Order 3: $C_3$ ($C_3^7.C_2^4:\GL(3,2)$)

Order 1: $C_1$ ($C_3^8:C_2^4.\GL(3,2)$)

Series

Derived series

$C_3^8:C_2^4.\GL(3,2)$

$\rhd$

$C_3^8.C_2^3:\GL(3,2)$

$\rhd$

$C_3^7.C_2^3:\GL(3,2)$

Chief series

$C_3^8:C_2^4.\GL(3,2)$

$\rhd$

$C_3^8:C_2^4$

$\rhd$

$C_3^8:C_2^3$

$\rhd$

$C_3^8$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^8:C_2^4.\GL(3,2)$

$\rhd$

$C_3^8.C_2^3:\GL(3,2)$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $240 \times 240$ character table is not available for this group.

Rational character table

The $234 \times 234$ rational character table is not available for this group.