Abstract group 115248.bg: $D_7\times C_7^3:S_4$

• Label: 115248.bg
• Order: \( 2^{4} \cdot 3 \cdot 7^{4} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \cdot 7^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \)
• Perm deg.: $28$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$D_7\times C_7^3:S_4$
Order:	\(115248\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{4} \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_7^3.(C_7\times A_4).C_6^2.C_2$, of order \(2074464\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7^{4} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_7$ x 4
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	7	14	21	28	42
Elements	1	1519	392	16464	2744	2400	44100	18816	12348	16464	115248
Conjugacy classes	1	5	1	2	1	119	185	27	3	6	350
Divisions	1	5	1	2	1	27	39	5	1	1	83
Autjugacy classes	1	5	1	2	1	15	27	3	1	1	57

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	$1026$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	12	36
Arbitrary	6	8	24

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid c^{14}=d^{14}=e^{7}=f^{7}=[a,f]=[c,f]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(1,3,7,13,12,17,19,14,2,5,11,16,21,6,8,4,9,15,20,18,10)(22,24)(23,25)(26,28) \!\cdots\! \rangle$
Transitive group:	42T1267				more information
Direct product:	$D_7$ $\, \times\, $ $(C_7^3:S_4)$
Semidirect product:	$(C_7\wr S_4)$ $\,\rtimes\,$ $C_2$	$(C_7^3\times D_7)$ $\,\rtimes\,$ $S_4$	$C_7^3$ $\,\rtimes\,$ $(D_7\times S_4)$	$(C_7^4:S_4)$ $\,\rtimes\,$ $C_2$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 193232 subgroups in 733 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_7\times C_7^3:S_4$
Commutator:	$G' \simeq$ $C_7\wr A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_7\times C_7^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_7^4$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $D_7\times C_7^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_7^4$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 115248: $D_7\times C_7^3:S_4$

Order 57624: $C_7\wr S_4$, $C_7^4:S_4$, $C_7^3:A_4\times D_7$

Order 38416: $C_7^4.C_2^3.C_2$

Order 28812: $C_7^4:D_6$, $C_7\wr A_4$

Order 19208: $D_7\times C_7^3:C_2^2$, $C_7^4:(C_2\times C_4)$, $C_7^4:D_4$, $C_7^4:D_4$, $C_7\times C_7^3:C_2^3$, $C_7^3:D_{28}$, $C_7\wr D_4$

Order 16464: $C_2\times C_7^3:S_4$

Order 14406: $C_7\times C_7^3:S_3$, $C_7^3:D_{21}$, $C_7^3:C_{42}$

Order 9604: $C_7\wr C_2^2$ x 2, $C_7^4:C_4$, $C_7\times D_7\times C_7:D_7$, $C_7\times C_7^3:C_2^2$, $C_7^4:C_2^2$, $C_7^2\times D_7^2$, $C_7\times C_7^3:C_2^2$, $C_7\wr C_4$

Order 8232: $C_7^3:S_4$ x 2, $C_2\times C_7^3:A_4$

Order 7203: $C_7^3:C_{21}$

Order 5488: $C_2\times C_7^3:D_4$, $D_7^3:C_2$

Order 4802: $C_7^2\wr C_2$ x 2, $C_7^3\times D_7$ x 2, $C_7\times C_7^3:C_2$

Order 4116: $C_7^3:D_6$, $C_7^3:D_6$, $C_7^3:A_4$

Order 2744: $C_7^3:D_4$ x 9, $D_7^3$ x 2, $C_{14}\times D_7^2$, $C_7\times D_{14}\times D_7$, $C_2\times C_7^3:C_4$, $C_2\times C_7^3:C_2^2$, $C_7^2:C_4\times D_7$, $D_7^2:C_{14}$, $C_7^2:D_{28}$

Order 2401: $C_7^4$

Order 2058: $C_7\wr S_3$ x 6, $C_7^2:D_{21}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^3:C_6$

Order 1372: $C_7\times D_7^2$ x 21, $C_7:D_7^2$ x 7, $C_7^3:C_4$ x 6, $D_{14}\times C_7^2$ x 3, $C_7:D_7^2$ x 3, $C_7^2:D_{14}$ x 2, $C_7^2:C_{28}$

Order 1029: $C_7\wr C_3$ x 5, $C_7^2:C_{21}$

Order 784: $D_7^2:C_2^2$, $D_{14}:D_{14}$

Order 686: $D_7\times C_7^2$ x 36, $C_7^2:C_{14}$ x 23, $C_7^2\times C_{14}$ x 2, $C_7^3:C_2$

Order 588: $C_{21}:D_{14}$, $C_7^2:D_6$

Order 392: $D_7\times D_{14}$ x 4, $D_7\wr C_2$ x 4, $C_{14}\times D_{14}$ x 2, $C_2\times C_7^2:C_4$, $C_{14}^2:C_2$, $C_{14}\wr C_2$, $C_7:D_{28}$, $D_{14}:D_7$, $C_{14}.D_{14}$

Order 343: $C_7^3$ x 27

Order 336: $D_7\times S_4$

Order 294: $C_7^2:S_3$ x 2, $C_7^2:C_6$, $C_7\times D_{21}$, $S_3\times C_7^2$, $D_7\times C_{21}$, $C_7^2:C_6$, $C_7:C_{42}$

Order 196: $C_7\times D_{14}$ x 28, $D_7^2$ x 19, $C_7:D_{14}$ x 3, $C_7^2:C_4$ x 2, $C_{14}^2$ x 2, $C_7^2:C_4$, $C_7:C_{28}$

Order 168: $A_4\times D_7$, $C_7:S_4$, $C_7\times S_4$

Order 147: $C_7:C_{21}$ x 5, $C_7\times C_{21}$, $C_7^2:C_3$

Order 112: $C_{14}:D_4$, $D_4\times D_7$

Order 98: $C_7\times D_7$ x 81, $C_7\times C_{14}$ x 30, $C_7:D_7$ x 15

Order 84: $S_3\times D_7$, $S_3\times C_{14}$, $C_7\times A_4$

Order 56: $C_7:D_4$ x 9, $C_2\times D_{14}$ x 4, $C_7\times D_4$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$, $C_2^2\times C_{14}$

Order 49: $C_7^2$ x 100

Order 48: $C_2\times S_4$

Order 42: $S_3\times C_7$ x 6, $C_{42}$, $D_{21}$, $C_3\times D_7$, $C_7:C_6$

Order 28: $D_{14}$ x 22, $C_2\times C_{14}$ x 14, $C_7:C_4$ x 6, $C_{28}$

Order 24: $S_4$ x 2, $C_2\times A_4$

Order 21: $C_{21}$ x 5, $C_7:C_3$

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 39, $D_7$ x 23

Order 12: $D_6$, $A_4$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 7: $C_7$ x 27

Order 6: $S_3$ x 2, $C_6$

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 115248: $D_7\times C_7^3:S_4$

Order 57624: $C_7\wr S_4$, $C_7^4:S_4$, $C_7^3:A_4\times D_7$

Order 38416: $C_7^4.C_2^3.C_2$

Order 28812: $C_7^4:D_6$, $C_7\wr A_4$

Order 19208: $D_7\times C_7^3:C_2^2$, $C_7^4:(C_2\times C_4)$, $C_7^4:D_4$, $C_7^4:D_4$, $C_7\times C_7^3:C_2^3$, $C_7^3:D_{28}$, $C_7\wr D_4$

Order 16464: $C_2\times C_7^3:S_4$

Order 14406: $C_7\times C_7^3:S_3$, $C_7^3:D_{21}$, $C_7^3:C_{42}$

Order 9604: $C_7\wr C_2^2$ x 2, $C_7^4:C_4$, $C_7\times D_7\times C_7:D_7$, $C_7\times C_7^3:C_2^2$, $C_7^4:C_2^2$, $C_7^2\times D_7^2$, $C_7\times C_7^3:C_2^2$, $C_7\wr C_4$

Order 8232: $C_7^3:S_4$ x 2, $C_2\times C_7^3:A_4$

Order 7203: $C_7^3:C_{21}$

Order 5488: $C_2\times C_7^3:D_4$, $D_7^3:C_2$

Order 4802: $C_7^2\wr C_2$ x 2, $C_7^3\times D_7$ x 2, $C_7\times C_7^3:C_2$

Order 4116: $C_7^3:D_6$, $C_7^3:D_6$, $C_7^3:A_4$

Order 2744: $C_7^3:D_4$ x 7, $D_7^3$ x 2, $C_{14}\times D_7^2$, $C_7\times D_{14}\times D_7$, $C_2\times C_7^3:C_4$, $C_2\times C_7^3:C_2^2$, $C_7^2:C_4\times D_7$, $D_7^2:C_{14}$, $C_7^2:D_{28}$

Order 2401: $C_7^4$

Order 2058: $C_7\wr S_3$ x 4, $C_7^2:D_{21}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^3:C_6$

Order 1372: $C_7\times D_7^2$ x 17, $C_7:D_7^2$ x 5, $C_7^3:C_4$ x 4, $D_{14}\times C_7^2$ x 3, $C_7:D_7^2$ x 3, $C_7^2:D_{14}$ x 2, $C_7^2:C_{28}$

Order 1029: $C_7\wr C_3$ x 3, $C_7^2:C_{21}$

Order 784: $D_7^2:C_2^2$, $D_{14}:D_{14}$

Order 686: $D_7\times C_7^2$ x 24, $C_7^2:C_{14}$ x 15, $C_7^2\times C_{14}$ x 2, $C_7^3:C_2$

Order 588: $C_{21}:D_{14}$, $C_7^2:D_6$

Order 392: $D_7\times D_{14}$ x 4, $D_7\wr C_2$ x 4, $C_{14}\times D_{14}$ x 2, $C_2\times C_7^2:C_4$, $C_{14}^2:C_2$, $C_{14}\wr C_2$, $C_7:D_{28}$, $D_{14}:D_7$, $C_{14}.D_{14}$

Order 343: $C_7^3$ x 15

Order 336: $D_7\times S_4$

Order 294: $C_7^2:S_3$ x 2, $C_7^2:C_6$, $C_7\times D_{21}$, $S_3\times C_7^2$, $D_7\times C_{21}$, $C_7^2:C_6$, $C_7:C_{42}$

Order 196: $C_7\times D_{14}$ x 24, $D_7^2$ x 15, $C_7:D_{14}$ x 3, $C_7^2:C_4$ x 2, $C_{14}^2$ x 2, $C_7^2:C_4$, $C_7:C_{28}$

Order 168: $A_4\times D_7$, $C_7:S_4$, $C_7\times S_4$

Order 147: $C_7:C_{21}$ x 3, $C_7\times C_{21}$, $C_7^2:C_3$

Order 112: $C_{14}:D_4$, $D_4\times D_7$

Order 98: $C_7\times D_7$ x 49, $C_7\times C_{14}$ x 20, $C_7:D_7$ x 9

Order 84: $S_3\times D_7$, $S_3\times C_{14}$, $C_7\times A_4$

Order 56: $C_7:D_4$ x 7, $C_2\times D_{14}$ x 4, $C_7\times D_4$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$, $C_2^2\times C_{14}$

Order 49: $C_7^2$ x 44

Order 48: $C_2\times S_4$

Order 42: $S_3\times C_7$ x 4, $C_{42}$, $D_{21}$, $C_3\times D_7$, $C_7:C_6$

Order 28: $D_{14}$ x 18, $C_2\times C_{14}$ x 12, $C_7:C_4$ x 4, $C_{28}$

Order 24: $S_4$ x 2, $C_2\times A_4$

Order 21: $C_{21}$ x 3, $C_7:C_3$

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 27, $D_7$ x 15

Order 12: $D_6$, $A_4$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 7: $C_7$ x 15

Order 6: $S_3$ x 2, $C_6$

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 115248: $D_7\times C_7^3:S_4$ ($C_1$)

Order 57624: $C_7\wr S_4$ ($C_2$), $C_7^4:S_4$ ($C_2$), $C_7^3:A_4\times D_7$ ($C_2$)

Order 28812: $C_7\wr A_4$ ($C_2^2$)

Order 19208: $D_7\times C_7^3:C_2^2$ ($S_3$)

Order 9604: $C_7\wr C_2^2$ ($D_6$)

Order 8232: $C_7^3:S_4$ ($D_7$)

Order 4802: $C_7^3\times D_7$ ($S_4$)

Order 4116: $C_7^3:A_4$ ($D_{14}$)

Order 2401: $C_7^4$ ($C_2\times S_4$)

Order 1372: $C_7:D_7^2$ ($S_3\times D_7$)

Order 343: $C_7^3$ ($D_7\times S_4$)

Order 14: $D_7$ ($C_7^3:S_4$)

Order 7: $C_7$ ($C_2\times C_7^3:S_4$)

Order 1: $C_1$ ($D_7\times C_7^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 115248: $D_7\times C_7^3:S_4$ ($C_1$)

Order 57624: $C_7\wr S_4$ ($C_2$), $C_7^4:S_4$ ($C_2$), $C_7^3:A_4\times D_7$ ($C_2$)

Order 28812: $C_7\wr A_4$ ($C_2^2$)

Order 19208: $D_7\times C_7^3:C_2^2$ ($S_3$)

Order 9604: $C_7\wr C_2^2$ ($D_6$)

Order 8232: $C_7^3:S_4$ ($D_7$)

Order 4802: $C_7^3\times D_7$ ($S_4$)

Order 4116: $C_7^3:A_4$ ($D_{14}$)

Order 2401: $C_7^4$ ($C_2\times S_4$)

Order 1372: $C_7:D_7^2$ ($S_3\times D_7$)

Order 343: $C_7^3$ ($D_7\times S_4$)

Order 14: $D_7$ ($C_7^3:S_4$)

Order 7: $C_7$ ($C_2\times C_7^3:S_4$)

Order 1: $C_1$ ($D_7\times C_7^3:S_4$)

Series

Derived series

$D_7\times C_7^3:S_4$

$\rhd$

$C_7\wr A_4$

$\rhd$

$C_7:D_7^2$

$\rhd$

$C_7^3$

$\rhd$

$C_1$

Chief series

$D_7\times C_7^3:S_4$

$\rhd$

$C_7^4:S_4$

$\rhd$

$C_7\wr A_4$

$\rhd$

$C_7\wr C_2^2$

$\rhd$

$C_7:D_7^2$

$\rhd$

$C_7^3$

$\rhd$

$C_1$

Lower central series

$D_7\times C_7^3:S_4$

$\rhd$

$C_7\wr A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $350 \times 350$ character table is not available for this group.

Rational character table

The $83 \times 83$ rational character table is not available for this group.