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

• Label: 5878656.d
• Order: \( 2^{7} \cdot 3^{8} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{8} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $21$
• Trans deg.: $21$
• Rank: $2$

Group information

Description:	$C_3^7.C_2^4:\GL(3,2)$
Order:	\(5878656\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 7 \)
Exponent:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Automorphism group:	$C_3^7.C_2^4:\GL(3,2)$, of order \(5878656\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 7, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	9	12	14	18	21
Elements	1	18063	56618	299376	1198638	279936	435456	1537704	839808	653184	559872	5878656
Conjugacy classes	1	7	15	6	53	2	8	20	2	4	2	120
Divisions	1	7	15	6	53	1	8	20	1	4	1	117
Autjugacy classes	1	7	15	6	53	2	8	20	2	4	2	120

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$21$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Permutation group:	Degree $21$ $\langle(1,7,16,3,9,17)(2,8,18)(4,14,21,6,15,19,5,13,20)(10,11,12), (1,2)(4,14,16,6,15,18)(5,13,17)(7,20,12,9,21,10,8,19,11)\rangle$
Transitive group:	21T136	24T23418	42T2487	42T2488	all 10
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:	$C_3^7$ . $(C_2^4:\GL(3,2))$	$(C_3^7:C_2^4)$ . $\PSL(2,7)$	$(C_3^7.C_2^3:\GL(3,2))$ . $C_2$	$(C_3^7:C_2^3)$ . $(C_2\times \GL(3,2))$	all 5
Aut. group:	$\Aut(C_3^7.C_2^3:\GL(3,2))$	$\Aut(C_3^7.C_2^4:\GL(3,2))$

Elements of the group are displayed as permutations of degree 21.

Homology

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

Subgroups

There are 7 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^7.C_2^4:\GL(3,2)$
Commutator:	$G' \simeq$ $C_3^7.C_2^3:\GL(3,2)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^7.C_2^4:\GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^7$	$G/\operatorname{Fit} \simeq$ $C_2^4:\GL(3,2)$
Radical:	$R \simeq$ $C_3^7:C_2^4$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_3^7$	$G/\operatorname{soc} \simeq$ $C_2^4:\GL(3,2)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_3^6.C_3$
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 128 (order at least 45927) are shown, as well as all normal subgroups of any index.

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

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

Order 839808: $C_3^7.C_2^4:S_4$, $C_3^7.C_2^4:S_4$

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

Order 419904: $C_3^4.S_3^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^4:A_4$, $C_3^7.C_2^4:A_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^4.C_6^3:S_4$

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

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

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

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

Order 139968: $C_3^6:C_2^3:S_4$ x 3, $C_3^6:C_2^3:S_4$ x 2, $C_3^5.D_6^2:C_4$ x 2, $C_3^2.S_3^5:C_2$ x 2, $C_3^7:C_2\wr C_2^2$ x 2, $C_3^7:C_2\wr C_2^2$ x 2, $C_3^7.C_2^4.C_2^2$, $C_3^6.C_2^3:S_4$, $C_3^6.C_2^3:S_4$, $C_3:S_3^3.S_3^3$, $C_3^7:(C_2^4:C_4)$, $C_3^6.C_2^4:A_4$, $C_3^7:C_2\wr C_2^2$, $C_3^7:C_2\wr C_2^2$, $C_3^7:C_2\wr C_2^2$, $C_3^7:C_2\wr C_2^2$

Order 122472: $C_3^7:F_8$, $C_3^6:\GL(3,2)$

Order 104976: $C_3^6.(S_3\times S_4)$ x 3, $C_3^6.(S_3\times S_4)$ x 2, $C_3^6.(A_4\times D_6)$ x 2, $C_3^6.(S_3\times S_4)$ x 2, $C_3^7.(C_2\times S_4)$ x 2, $C_3^6.(S_3\times S_4)$ x 2, $C_3^6.(C_6\times S_4)$ x 2, $C_3^5.A_4.C_3.D_6$, $C_3^4.C_6^2.S_3^2$, $C_3^7.C_2^2:A_4$, $C_3\wr S_4.S_3$, $C_3\wr S_4.C_6$, $C_3\wr A_4.D_6$, $C_3\wr A_4.D_6$, $C_3\wr A_4.D_6$, $C_3\wr A_4.D_6$, $C_3^7.(C_2\times \SL(2,3))$

Order 93312: $C_3^4.D_6^2:D_4$

Order 91854: $C_3^6.C_{21}:C_6$

Order 69984: $C_3^5:D_6:S_4$ x 3, $C_3^6:C_2^3:A_4$ x 3, $C_3^4.S_3^3:C_4$ x 3, $C_3^7:(C_2^2\times D_4)$ x 3, $C_3^7:C_2^2\wr C_2$ x 3, $C_3^7:C_2^2\wr C_2$ x 3, $C_3^6.C_2^4.S_3$ x 2, $C_3^2:S_3^2.S_3^3$ x 2, $C_3^5:D_6.D_{12}$ x 2, $C_3^4:C_6^2.D_{12}$ x 2, $C_3^7:C_2^2.D_4$ x 2, $C_3^5:D_6.D_{12}$ x 2, $C_3^6.C_2^3:C_{12}$ x 2, $C_3^7:C_2^2\wr C_2$ x 2, $C_3^7:C_2^2\wr C_2$ x 2, $C_3^6.(C_2^2\times S_4)$ x 2, $C_3^5.C_6^2.C_2^3$, $C_3^7.C_2^2\wr C_2$, $C_3^7.C_2^2\wr C_2$, $C_3^6.C_2^4:S_3$, $C_3^7.C_2^5$, $C_3^7.C_2^4.C_2$, $C_3^5.C_6^2.C_2^3$, $C_3^6.C_2^3:D_6$, $C_3^7.C_2^4.C_2$, $C_3^7.C_2^4.C_2$, $C_3^7.C_2^3.C_2^2$, $C_3^5:D_6:S_4$, $C_3^5:(D_6\times S_4)$, $(C_3:S_3)^3.A_4$, $C_3^2:S_3^2.S_3^3$, $C_3\wr A_4:C_2^3$, $C_3^2:S_3^2.S_3^3$, $C_3^4:C_6.D_6^2$, $C_3^4:C_6.D_6^2$, $C_3^4.C_6^3:C_4$, $C_3^4.C_6^3:C_4$

Order 52488: $C_3^6.(S_3\times A_4)$ x 3, $C_3^7.S_4$ x 3, $C_3^7.S_4$ x 3, $C_3^7.A_4.C_2$ x 2, $C_3^5.C_6^2.S_3$ x 2, $C_3^7.S_4$ x 2, $C_3^7.S_4$ x 2, $C_3^6.(S_3\times A_4)$ x 2, $C_3^6.(C_6\times A_4)$ x 2, $C_3^4.C_6^2:C_3.C_6$, $C_3^3.C_6^2.C_3^3.C_2$, $C_3^5.S_3^3$, $C_3^7.S_4$, $C_3^7.S_4$, $C_3\wr A_4.S_3$, $C_3\wr A_4.C_6$, $C_3^7.\SL(2,3)$

Order 46656: $C_3^6:C_2\wr C_2^2$ x 4, $C_3^6:C_2\wr C_2^2$ x 4, $C_3^4.D_6^2:C_4$ x 2, $(C_3:S_3)^3.D_4$ x 2, $C_3^6.C_2^4.C_2^2$, $C_3^6.C_2^4.C_2^2$, $C_3^6.C_2.C_2^5$, $(C_3:S_3)^3.D_4$

Order 45927: $C_3^6.C_{21}:C_3$

Order 34992: $C_3^7:C_2^4$

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

Order 6561: $C_3\times C_3^6.C_3$

Order 4374: $C_3^7.C_2$

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

Order 2187: $C_3^7$

Order 128: $C_2^4:D_4$

Order 7: $C_7$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 45927) are shown, as well as all normal subgroups of any index.

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

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

Order 839808: $C_3^7.C_2^4:S_4$, $C_3^7.C_2^4:S_4$

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

Order 419904: $C_3^4.S_3^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^4:A_4$, $C_3^7.C_2^4:A_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^7.C_2^3:S_4$, $C_3^4.C_6^3:S_4$

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

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

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

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

Order 139968: $C_3^6:C_2^3:S_4$ x 3, $C_3^6:C_2^3:S_4$ x 2, $C_3^5.D_6^2:C_4$ x 2, $C_3^2.S_3^5:C_2$ x 2, $C_3^7:C_2\wr C_2^2$ x 2, $C_3^7:C_2\wr C_2^2$ x 2, $C_3^7.C_2^4.C_2^2$, $C_3^6.C_2^3:S_4$, $C_3^6.C_2^3:S_4$, $C_3:S_3^3.S_3^3$, $C_3^7:(C_2^4:C_4)$, $C_3^6.C_2^4:A_4$, $C_3^7:C_2\wr C_2^2$, $C_3^7:C_2\wr C_2^2$, $C_3^7:C_2\wr C_2^2$, $C_3^7:C_2\wr C_2^2$

Order 122472: $C_3^7:F_8$, $C_3^6:\GL(3,2)$

Order 104976: $C_3^6.(S_3\times S_4)$ x 3, $C_3^6.(S_3\times S_4)$ x 2, $C_3^6.(A_4\times D_6)$ x 2, $C_3^6.(S_3\times S_4)$ x 2, $C_3^7.(C_2\times S_4)$ x 2, $C_3^6.(S_3\times S_4)$ x 2, $C_3^6.(C_6\times S_4)$ x 2, $C_3^5.A_4.C_3.D_6$, $C_3^4.C_6^2.S_3^2$, $C_3^7.C_2^2:A_4$, $C_3\wr S_4.S_3$, $C_3\wr S_4.C_6$, $C_3\wr A_4.D_6$, $C_3\wr A_4.D_6$, $C_3\wr A_4.D_6$, $C_3\wr A_4.D_6$, $C_3^7.(C_2\times \SL(2,3))$

Order 93312: $C_3^4.D_6^2:D_4$

Order 91854: $C_3^6.C_{21}:C_6$

Order 69984: $C_3^5:D_6:S_4$ x 3, $C_3^6:C_2^3:A_4$ x 3, $C_3^4.S_3^3:C_4$ x 3, $C_3^7:(C_2^2\times D_4)$ x 3, $C_3^7:C_2^2\wr C_2$ x 3, $C_3^7:C_2^2\wr C_2$ x 3, $C_3^6.C_2^4.S_3$ x 2, $C_3^2:S_3^2.S_3^3$ x 2, $C_3^5:D_6.D_{12}$ x 2, $C_3^4:C_6^2.D_{12}$ x 2, $C_3^7:C_2^2.D_4$ x 2, $C_3^5:D_6.D_{12}$ x 2, $C_3^6.C_2^3:C_{12}$ x 2, $C_3^7:C_2^2\wr C_2$ x 2, $C_3^7:C_2^2\wr C_2$ x 2, $C_3^6.(C_2^2\times S_4)$ x 2, $C_3^5.C_6^2.C_2^3$, $C_3^7.C_2^2\wr C_2$, $C_3^7.C_2^2\wr C_2$, $C_3^6.C_2^4:S_3$, $C_3^7.C_2^5$, $C_3^7.C_2^4.C_2$, $C_3^5.C_6^2.C_2^3$, $C_3^6.C_2^3:D_6$, $C_3^7.C_2^4.C_2$, $C_3^7.C_2^4.C_2$, $C_3^7.C_2^3.C_2^2$, $C_3^5:D_6:S_4$, $C_3^5:(D_6\times S_4)$, $(C_3:S_3)^3.A_4$, $C_3^2:S_3^2.S_3^3$, $C_3\wr A_4:C_2^3$, $C_3^2:S_3^2.S_3^3$, $C_3^4:C_6.D_6^2$, $C_3^4:C_6.D_6^2$, $C_3^4.C_6^3:C_4$, $C_3^4.C_6^3:C_4$

Order 52488: $C_3^6.(S_3\times A_4)$ x 3, $C_3^7.S_4$ x 3, $C_3^7.S_4$ x 3, $C_3^7.A_4.C_2$ x 2, $C_3^5.C_6^2.S_3$ x 2, $C_3^7.S_4$ x 2, $C_3^7.S_4$ x 2, $C_3^6.(S_3\times A_4)$ x 2, $C_3^6.(C_6\times A_4)$ x 2, $C_3^4.C_6^2:C_3.C_6$, $C_3^3.C_6^2.C_3^3.C_2$, $C_3^5.S_3^3$, $C_3^7.S_4$, $C_3^7.S_4$, $C_3\wr A_4.S_3$, $C_3\wr A_4.C_6$, $C_3^7.\SL(2,3)$

Order 46656: $C_3^6:C_2\wr C_2^2$ x 4, $C_3^6:C_2\wr C_2^2$ x 4, $C_3^4.D_6^2:C_4$ x 2, $(C_3:S_3)^3.D_4$ x 2, $C_3^6.C_2^4.C_2^2$, $C_3^6.C_2^4.C_2^2$, $C_3^6.C_2.C_2^5$, $(C_3:S_3)^3.D_4$

Order 45927: $C_3^6.C_{21}:C_3$

Order 34992: $C_3^7:C_2^4$

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

Order 6561: $C_3\times C_3^6.C_3$

Order 4374: $C_3^7.C_2$

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

Order 2187: $C_3^7$

Order 128: $C_2^4:D_4$

Order 7: $C_7$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5878656: $C_3^7.C_2^4:\GL(3,2)$ ($C_1$)

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

Order 34992: $C_3^7:C_2^4$ ($\PSL(2,7)$)

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

Order 4374: $C_3^7.C_2$ ($C_2^3:\GL(3,2)$)

Order 2187: $C_3^7$ ($C_2^4:\GL(3,2)$)

Order 1: $C_1$ ($C_3^7.C_2^4:\GL(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5878656: $C_3^7.C_2^4:\GL(3,2)$ ($C_1$)

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

Order 34992: $C_3^7:C_2^4$ ($\PSL(2,7)$)

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

Order 4374: $C_3^7.C_2$ ($C_2^3:\GL(3,2)$)

Order 2187: $C_3^7$ ($C_2^4:\GL(3,2)$)

Order 1: $C_1$ ($C_3^7.C_2^4:\GL(3,2)$)

Series

Derived series	$C_3^7.C_2^4:\GL(3,2)$	$\rhd$	$C_3^7.C_2^3:\GL(3,2)$
Chief series	$C_3^7.C_2^4:\GL(3,2)$	$\rhd$	$C_3^7:C_2^4$	$\rhd$	$C_3^7:C_2^3$	$\rhd$	$C_3^7$	$\rhd$	$C_1$
Lower central series	$C_3^7.C_2^4:\GL(3,2)$	$\rhd$	$C_3^7.C_2^3:\GL(3,2)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

The $120 \times 120$ character table is not available for this group.

Rational character table

The $117 \times 117$ rational character table is not available for this group.