Abstract group 16464.bz: $C_2\times C_7^3:S_4$

• Label: 16464.bz
• Order: \( 2^{4} \cdot 3 \cdot 7^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{2} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $23$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$C_2\times C_7^3:S_4$
Order:	\(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$D_7^3:(C_6\times S_3)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_7$ x 3
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	7	14	21	42
Elements	1	379	392	4116	392	342	6138	2352	2352	16464
Conjugacy classes	1	5	1	2	1	29	89	6	6	140
Divisions	1	5	1	2	1	7	19	1	1	38
Autjugacy classes	1	4	1	1	1	7	14	1	1	31

Dimension	1	2	3	4	6	8	12	18	24	36	48	72
Irr. complex chars.	4	2	4	24	24	12	60	0	10	0	0	0	140
Irr. rational chars.	4	2	4	0	0	0	0	4	4	6	4	10	38

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	$513$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	6	18
Arbitrary	4	6	18

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid c^{2}=d^{14}=e^{7}=f^{7}=[a,f]=[c,f]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $23$ $\langle(1,3,7,13,19,17,8)(2,5,9,16,14,20,12,15,21,4,6,11,18,10)(22,23), (1,2,4,3,6,10,7,12,16,13,9,15,19,18,11,17,21,5,8,14,20)(22,23)\rangle$
Transitive group:	42T703	42T704			more information
Direct product:	$C_2$ $\, \times\, $ $(C_7^3:S_4)$
Semidirect product:	$C_7^3$ $\,\rtimes\,$ $(C_2\times S_4)$	$(C_7:D_7^2)$ $\,\rtimes\,$ $D_6$	$(C_7^2\times C_{14})$ $\,\rtimes\,$ $S_4$	$(C_7^3:A_4)$ $\,\rtimes\,$ $C_2^2$	all 5
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 21604 subgroups in 210 conjugacy classes, 11 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_7^3:S_4$
Commutator:	$G' \simeq$ $C_7^3:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times C_7^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_7^2\times C_{14}$	$G/\operatorname{Fit} \simeq$ $S_4$
Radical:	$R \simeq$ $C_2\times C_7^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_7^2\times C_{14}$	$G/\operatorname{soc} \simeq$ $S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

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

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

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

Order 2744: $C_7^3:D_4$ x 4, $C_{14}\times D_7^2$, $C_2\times C_7^3:C_4$, $C_2\times C_7^3:C_2^2$

Order 2058: $C_7\wr S_3$ x 2, $C_7^2:C_{42}$

Order 1372: $C_7\times D_7^2$ x 3, $C_7:D_7^2$ x 2, $C_7^3:C_4$ x 2, $C_7^2:D_{14}$, $D_{14}\times C_7^2$

Order 1029: $C_7\wr C_3$

Order 784: $D_7^2:C_2^2$

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

Order 588: $C_7^2:D_6$

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

Order 343: $C_7^3$

Order 294: $C_7^2:S_3$ x 2, $C_7^2:C_6$, $C_7:C_{42}$

Order 196: $C_7\times D_{14}$ x 12, $D_7^2$ x 6, $C_7^2:C_4$ x 2, $C_{14}^2$, $C_7:D_{14}$

Order 147: $C_7^2:C_3$, $C_7:C_{21}$

Order 112: $C_{14}:D_4$

Order 98: $C_7\times D_7$ x 16, $C_7\times C_{14}$ x 9, $C_7:D_7$ x 2

Order 84: $S_3\times C_{14}$

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

Order 49: $C_7^2$ x 7

Order 48: $C_2\times S_4$

Order 42: $S_3\times C_7$ x 2, $C_{42}$, $C_7:C_6$

Order 28: $D_{14}$ x 11, $C_2\times C_{14}$ x 9, $C_7:C_4$ x 2

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

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

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 19, $D_7$ x 8

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 7

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

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

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

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

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

Order 2058: $C_7\wr S_3$, $C_7^2:C_{42}$

Order 1372: $C_7:D_7^2$ x 2, $C_7\times D_7^2$ x 2, $C_7^2:D_{14}$, $D_{14}\times C_7^2$, $C_7^3:C_4$

Order 1029: $C_7\wr C_3$

Order 784: $D_7^2:C_2^2$

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

Order 588: $C_7^2:D_6$

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

Order 343: $C_7^3$

Order 294: $C_7^2:S_3$, $C_7^2:C_6$, $C_7:C_{42}$

Order 196: $C_7\times D_{14}$ x 10, $D_7^2$ x 5, $C_7^2:C_4$, $C_{14}^2$, $C_7:D_{14}$

Order 147: $C_7^2:C_3$, $C_7:C_{21}$

Order 112: $C_{14}:D_4$

Order 98: $C_7\times D_7$ x 11, $C_7\times C_{14}$ x 8, $C_7:D_7$ x 2

Order 84: $S_3\times C_{14}$

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

Order 49: $C_7^2$ x 7

Order 48: $C_2\times S_4$

Order 42: $C_{42}$, $S_3\times C_7$, $C_7:C_6$

Order 28: $D_{14}$ x 9, $C_2\times C_{14}$ x 8, $C_7:C_4$

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

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

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 14, $D_7$ x 7

Order 12: $D_6$, $A_4$

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

Order 7: $C_7$ x 7

Order 6: $C_6$, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16464: $C_2\times C_7^3:S_4$ ($C_1$)

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

Order 4116: $C_7^3:A_4$ ($C_2^2$)

Order 2744: $C_2\times C_7^3:C_2^2$ ($S_3$)

Order 1372: $C_7:D_7^2$ ($D_6$)

Order 686: $C_7^2\times C_{14}$ ($S_4$)

Order 343: $C_7^3$ ($C_2\times S_4$)

Order 2: $C_2$ ($C_7^3:S_4$)

Order 1: $C_1$ ($C_2\times C_7^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16464: $C_2\times C_7^3:S_4$ ($C_1$)

Order 8232: $C_2\times C_7^3:A_4$ ($C_2$), $C_7^3:S_4$ ($C_2$)

Order 4116: $C_7^3:A_4$ ($C_2^2$)

Order 2744: $C_2\times C_7^3:C_2^2$ ($S_3$)

Order 1372: $C_7:D_7^2$ ($D_6$)

Order 686: $C_7^2\times C_{14}$ ($S_4$)

Order 343: $C_7^3$ ($C_2\times S_4$)

Order 2: $C_2$ ($C_7^3:S_4$)

Order 1: $C_1$ ($C_2\times C_7^3:S_4$)

Series

Derived series

$C_2\times C_7^3:S_4$

$\rhd$

$C_7^3:A_4$

$\rhd$

$C_7:D_7^2$

$\rhd$

$C_7^3$

$\rhd$

$C_1$

Chief series

$C_2\times C_7^3:S_4$

$\rhd$

$C_7^3:S_4$

$\rhd$

$C_7^3:A_4$

$\rhd$

$C_7:D_7^2$

$\rhd$

$C_7^3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_7^3:S_4$

$\rhd$

$C_7^3:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 5 larger groups in the database.

Character theory

Complex character table

See the $140 \times 140$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.