Abstract group 8232.ba: $C_7^2:(C_2^2\times F_7)$

• Label: 8232.ba
• Order: \( 2^{3} \cdot 3 \cdot 7^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{2} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $51$
• Trans deg.: $98$
• Rank: $3$

Group information

Description:	$C_7^2:(C_2^2\times F_7)$
Order:	\(8232\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{3} \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$\He_7.C_6^2.C_2^3$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \)
Composition factors:	$C_2$ x 3, $C_3$, $C_7$ x 3
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	295	98	3038	342	2106	588	1764	8232
Conjugacy classes	1	7	2	14	8	18	6	18	74
Divisions	1	7	1	7	4	10	1	3	34
Autjugacy classes	1	3	1	4	3	5	1	2	20

Dimension	1	2	6	12	14	36	42	84
Irr. complex chars.	24	0	8	6	36	0	0	0	74
Irr. rational chars.	8	8	8	0	0	2	4	4	34

Minimal presentations

Permutation degree:	$51$
Transitive degree:	$98$
Rank:	$3$
Inequivalent generating triples:	$1712256$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	14	42
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d \mid b^{14}=c^{7}=d^{14}=[a,d]=[c,d]=1, a^{6}=d^{2}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $51$ $\langle(1,2,7,25,40,23,48)(3,8,26,33,46,28,44)(4,14,21,24,39,15,41)(5,18,32,10,27,47,35) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 6 & 1 & 3 & 3 \\ 3 & 2 & 4 & 3 \\ 1 & 5 & 3 & 6 \\ 5 & 1 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 4 & 1 & 0 & 0 \\ 5 & 0 & 1 & 0 \\ 0 & 2 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 5 & 6 \\ 3 & 0 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 5 & 2 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 1 & 3 \\ 3 & 6 & 5 & 1 \\ 3 & 5 & 3 & 6 \\ 6 & 3 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 1 & 5 \\ 0 & 2 & 1 & 4 \\ 6 & 0 & 1 & 4 \\ 1 & 3 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 3 & 2 \\ 1 & 5 & 1 & 3 \\ 3 & 5 & 4 & 2 \\ 6 & 3 & 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_2$ $\, \times\, $ $(C_7^2:(C_2\times F_7))$
Semidirect product:	$(C_7:D_{14})$ $\,\rtimes\,$ $F_7$	$(C_{14}.D_7^2)$ $\,\rtimes\,$ $C_3$	$(C_7^2:D_{14})$ $\,\rtimes\,$ $C_6$	$(C_7^2:D_{14})$ $\,\rtimes\,$ $C_6$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{14}$ . $(D_7:F_7)$	$C_7$ . $(D_{14}:F_7)$			more information

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 10460 subgroups in 234 conjugacy classes, 46 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_7^2:(C_2\times F_7)$
Commutator:	$G' \simeq$ $\He_7$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_7$	$G/\Phi \simeq$ $D_{14}:F_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times \He_7$	$G/\operatorname{Fit} \simeq$ $C_2\times C_6$
Radical:	$R \simeq$ $C_7^2:(C_2^2\times F_7)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $D_7:F_7$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $\He_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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

Classes of subgroups up to conjugation

Order 8232: $C_7^2:(C_2^2\times F_7)$

Order 4116: $C_7^2:(C_2\times F_7)$ x 4, $C_2\times C_7^2:F_7$ x 2, $C_2\times C_7^2:F_7$

Order 2744: $C_{14}.D_7^2$

Order 2058: $C_7^2:F_7$ x 4, $C_7^2:F_7$ x 2, $\He_7:C_6$

Order 1372: $C_7^2:D_{14}$ x 4, $C_7^2:D_{14}$ x 2, $C_7^2:D_{14}$

Order 1176: $D_{14}\times F_7$ x 2

Order 1029: $\He_7:C_3$

Order 686: $C_7^2:C_{14}$ x 4, $C_7^2:D_7$ x 2, $C_2\times \He_7$

Order 588: $D_7\times F_7$ x 8, $C_{14}:F_7$ x 2, $C_7:C_3\times D_{14}$ x 2, $C_{14}\times F_7$ x 2

Order 392: $D_7\times D_{14}$ x 2

Order 343: $\He_7$

Order 294: $C_7^2:C_6$ x 4, $C_7\times F_7$ x 4, $C_7:F_7$ x 4, $C_7:C_{42}$ x 2

Order 196: $D_7^2$ x 8, $C_7\times D_{14}$ x 5, $C_7:D_{14}$ x 2

Order 168: $C_2^2\times F_7$ x 2, $C_6\times D_{14}$

Order 147: $C_7:C_{21}$ x 2

Order 98: $C_7\times D_7$ x 10, $C_7:D_7$ x 4, $C_7\times C_{14}$ x 3

Order 84: $C_2\times F_7$ x 12, $C_3\times D_{14}$ x 6, $C_{14}:C_6$ x 2, $C_2\times C_{42}$

Order 56: $C_2\times D_{14}$ x 3

Order 49: $C_7^2$ x 3

Order 42: $F_7$ x 8, $C_7:C_6$ x 6, $C_3\times D_7$ x 4, $C_{42}$ x 3

Order 28: $D_{14}$ x 19, $C_2\times C_{14}$ x 3

Order 24: $C_2^2\times C_6$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 14: $D_7$ x 14, $C_{14}$ x 10

Order 12: $C_2\times C_6$ x 7

Order 8: $C_2^3$

Order 7: $C_7$ x 4

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 8232: $C_7^2:(C_2^2\times F_7)$

Order 4116: $C_7^2:(C_2\times F_7)$, $C_2\times C_7^2:F_7$, $C_2\times C_7^2:F_7$

Order 2744: $C_{14}.D_7^2$

Order 2058: $\He_7:C_6$, $C_7^2:F_7$, $C_7^2:F_7$

Order 1372: $C_7^2:D_{14}$, $C_7^2:D_{14}$, $C_7^2:D_{14}$

Order 1176: $D_{14}\times F_7$

Order 1029: $\He_7:C_3$

Order 686: $C_7^2:D_7$, $C_7^2:C_{14}$, $C_2\times \He_7$

Order 588: $C_{14}:F_7$, $C_7:C_3\times D_{14}$, $C_{14}\times F_7$, $D_7\times F_7$

Order 392: $D_7\times D_{14}$

Order 343: $\He_7$

Order 294: $C_7^2:C_6$, $C_7\times F_7$, $C_7:C_{42}$, $C_7:F_7$

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

Order 168: $C_6\times D_{14}$, $C_2^2\times F_7$

Order 147: $C_7:C_{21}$

Order 98: $C_7\times D_7$ x 3, $C_7\times C_{14}$ x 2, $C_7:D_7$

Order 84: $C_2\times F_7$ x 3, $C_3\times D_{14}$ x 2, $C_{14}:C_6$, $C_2\times C_{42}$

Order 56: $C_2\times D_{14}$ x 2

Order 49: $C_7^2$ x 2

Order 42: $C_{42}$ x 2, $C_7:C_6$ x 2, $F_7$ x 2, $C_3\times D_7$

Order 28: $D_{14}$ x 6, $C_2\times C_{14}$ x 2

Order 24: $C_2^2\times C_6$

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

Order 14: $C_{14}$ x 5, $D_7$ x 4

Order 12: $C_2\times C_6$ x 3

Order 8: $C_2^3$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 8232: $C_7^2:(C_2^2\times F_7)$ ($C_1$)

Order 4116: $C_7^2:(C_2\times F_7)$ ($C_2$) x 4, $C_2\times C_7^2:F_7$ ($C_2$) x 2, $C_2\times C_7^2:F_7$ ($C_2$)

Order 2744: $C_{14}.D_7^2$ ($C_3$)

Order 2058: $C_7^2:F_7$ ($C_2^2$) x 4, $C_7^2:F_7$ ($C_2^2$) x 2, $\He_7:C_6$ ($C_2^2$)

Order 1372: $C_7^2:D_{14}$ ($C_6$) x 4, $C_7^2:D_{14}$ ($C_6$) x 2, $C_7^2:D_{14}$ ($C_6$)

Order 1029: $\He_7:C_3$ ($C_2^3$)

Order 686: $C_7^2:C_{14}$ ($C_2\times C_6$) x 4, $C_7^2:D_7$ ($C_2\times C_6$) x 2, $C_2\times \He_7$ ($C_2\times C_6$)

Order 343: $\He_7$ ($C_2^2\times C_6$)

Order 196: $C_7:D_{14}$ ($F_7$) x 2

Order 98: $C_7:D_7$ ($C_2\times F_7$) x 4, $C_7\times C_{14}$ ($C_2\times F_7$) x 2

Order 49: $C_7^2$ ($C_2^2\times F_7$) x 2

Order 14: $C_{14}$ ($D_7:F_7$)

Order 7: $C_7$ ($D_{14}:F_7$)

Order 2: $C_2$ ($C_7^2:(C_2\times F_7)$)

Order 1: $C_1$ ($C_7^2:(C_2^2\times F_7)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8232: $C_7^2:(C_2^2\times F_7)$ ($C_1$)

Order 4116: $C_7^2:(C_2\times F_7)$ ($C_2$), $C_2\times C_7^2:F_7$ ($C_2$), $C_2\times C_7^2:F_7$ ($C_2$)

Order 2744: $C_{14}.D_7^2$ ($C_3$)

Order 2058: $\He_7:C_6$ ($C_2^2$), $C_7^2:F_7$ ($C_2^2$), $C_7^2:F_7$ ($C_2^2$)

Order 1372: $C_7^2:D_{14}$ ($C_6$), $C_7^2:D_{14}$ ($C_6$), $C_7^2:D_{14}$ ($C_6$)

Order 1029: $\He_7:C_3$ ($C_2^3$)

Order 686: $C_7^2:D_7$ ($C_2\times C_6$), $C_7^2:C_{14}$ ($C_2\times C_6$), $C_2\times \He_7$ ($C_2\times C_6$)

Order 343: $\He_7$ ($C_2^2\times C_6$)

Order 196: $C_7:D_{14}$ ($F_7$)

Order 98: $C_7\times C_{14}$ ($C_2\times F_7$), $C_7:D_7$ ($C_2\times F_7$)

Order 49: $C_7^2$ ($C_2^2\times F_7$)

Order 14: $C_{14}$ ($D_7:F_7$)

Order 7: $C_7$ ($D_{14}:F_7$)

Order 2: $C_2$ ($C_7^2:(C_2\times F_7)$)

Order 1: $C_1$ ($C_7^2:(C_2^2\times F_7)$)

Series

Derived series

$C_7^2:(C_2^2\times F_7)$

$\rhd$

$C_7^2:(C_2^2\times F_7)$

$\rhd$

$\He_7$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_7^2:(C_2^2\times F_7)$

$\rhd$

$C_7^2:(C_2^2\times F_7)$

$\rhd$

$C_2\times C_7^2:F_7$

$\rhd$

$C_2\times C_7^2:F_7$

$\rhd$

$C_7^2:D_{14}$

$\rhd$

$C_7^2:D_{14}$

$\rhd$

$C_2\times \He_7$

$\rhd$

$C_2\times \He_7$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_7^2:(C_2^2\times F_7)$

$\rhd$

$C_7^2:(C_2^2\times F_7)$

$\rhd$

$\He_7$

$\rhd$

$\He_7$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

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 $74 \times 74$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $34 \times 34$ rational character table.