Abstract group 12348.bp: $C_7^3:C_6^2$

• Label: 12348.bp
• Order: \( 2^{2} \cdot 3^{2} \cdot 7^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{4} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $23$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$C_7^3:C_6^2$
Order:	\(12348\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{3} \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$C_2.F_7\wr S_3$, of order \(889056\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 7^{3} \)
Composition factors:	$C_2$ x 2, $C_3$ x 2, $C_7$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	687	980	6468	342	342	1764	1764	12348
Conjugacy classes	1	3	8	24	21	21	6	6	90
Divisions	1	3	4	12	21	21	3	3	68
Autjugacy classes	1	2	3	6	3	3	1	1	20

Dimension	1	2	6	12	18
Irr. complex chars.	36	0	18	0	36	90
Irr. rational chars.	4	16	6	6	36	68

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d \mid a^{6}=b^{42}=c^{7}=d^{7}=[c,d]=1, b^{a}=b^{19}c^{2}d^{4}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $23$ $\langle(1,2,3,5,4,7)(8,9)(10,11)(12,13)(15,16,18,17,19,21)(22,23), (2,4,7)(3,6,5)(8,9,10)(11,12,14)(15,17,19,18,20,21,16)(22,23)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 6 & 2 & 5 \\ 4 & 5 & 0 & 2 \\ 1 & 3 & 4 & 1 \\ 6 & 1 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 5 & 0 & 5 & 5 \\ 1 & 4 & 6 & 5 \\ 1 & 6 & 3 & 0 \\ 4 & 1 & 6 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 5 & 1 & 2 \\ 1 & 6 & 2 & 5 \\ 5 & 6 & 4 & 3 \\ 1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 3 & 3 \\ 1 & 3 & 0 & 5 \\ 1 & 2 & 0 & 2 \\ 4 & 5 & 5 & 5 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 3 & 5 & 3 \\ 1 & 6 & 6 & 5 \\ 5 & 4 & 3 & 4 \\ 1 & 5 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 3 & 2 \\ 1 & 5 & 1 & 3 \\ 3 & 5 & 4 & 2 \\ 6 & 3 & 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Transitive group:	42T627				more information
Direct product:	$C_2$ $\, \times\, $ $(C_7^3:(C_3\times C_6))$
Semidirect product:	$C_7^3$ $\,\rtimes\,$ $C_6^2$	$(C_7^3:C_6)$ $\,\rtimes\,$ $C_6$	$(C_7^3:C_6)$ $\,\rtimes\,$ $C_6$	$(C_7^3:C_6)$ $\,\rtimes\,$ $C_6$	all 20
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:	$C_{6}^{2} \simeq C_{2}^{2} \times C_{3}^{2}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 36456 subgroups in 600 conjugacy classes, 50 normal (11 characteristic).

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

Special subgroups

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 12348: $C_7^3:C_6^2$

Order 6174: $C_7^3:(C_3\times C_6)$ x 2, $C_2\times C_7^3:C_3^2$

Order 4116: $C_2\times C_7^3:C_6$ x 3, $C_2\times C_7^3:C_6$

Order 3087: $C_7^3:C_3^2$

Order 2058: $C_7^3:C_6$ x 6, $C_7^2:C_{42}$ x 3, $C_7^3:C_6$ x 2, $C_7^3:C_6$

Order 1764: $C_7^2:C_6^2$ x 3

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

Order 1029: $C_7\wr C_3$ x 3, $C_7^3:C_3$

Order 882: $C_7:(C_3\times F_7)$ x 6, $C_2\times C_7^2:C_3^2$ x 3

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

Order 588: $C_{14}:F_7$ x 21, $C_{14}:F_7$ x 6, $C_{14}:F_7$ x 3

Order 441: $C_7^2:C_3^2$ x 3

Order 343: $C_7^3$

Order 294: $C_7:F_7$ x 42, $C_7^2:C_6$ x 21, $C_7:F_7$ x 12, $C_7:C_{42}$ x 6, $C_7:F_7$ x 6, $C_7^2:C_6$ x 3

Order 252: $C_6\times F_7$ x 3

Order 196: $C_7:D_{14}$ x 21

Order 147: $C_7^2:C_3$ x 21, $C_7:C_{21}$ x 6, $C_7^2:C_3$ x 3

Order 126: $C_3\times F_7$ x 6, $C_{21}:C_6$ x 3

Order 98: $C_7:D_7$ x 42, $C_7\times C_{14}$ x 21

Order 84: $C_2\times F_7$ x 27, $C_3\times D_{14}$ x 3

Order 63: $C_{21}:C_3$ x 3

Order 49: $C_7^2$ x 21

Order 42: $F_7$ x 54, $C_7:C_6$ x 27, $C_3\times D_7$ x 6, $C_{42}$ x 3

Order 36: $C_6^2$

Order 28: $D_{14}$ x 21

Order 21: $C_7:C_3$ x 27, $C_{21}$ x 3

Order 18: $C_3\times C_6$ x 3

Order 14: $D_7$ x 42, $C_{14}$ x 21

Order 12: $C_2\times C_6$ x 4

Order 9: $C_3^2$

Order 7: $C_7$ x 21

Order 6: $C_6$ x 12

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12348: $C_7^3:C_6^2$

Order 6174: $C_7^3:(C_3\times C_6)$, $C_2\times C_7^3:C_3^2$

Order 4116: $C_2\times C_7^3:C_6$, $C_2\times C_7^3:C_6$

Order 3087: $C_7^3:C_3^2$

Order 2058: $C_7^3:C_6$, $C_7^3:C_6$, $C_7^2:C_{42}$, $C_7^3:C_6$

Order 1764: $C_7^2:C_6^2$

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

Order 1029: $C_7^3:C_3$, $C_7\wr C_3$

Order 882: $C_2\times C_7^2:C_3^2$, $C_7:(C_3\times F_7)$

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

Order 588: $C_{14}:F_7$ x 3, $C_{14}:F_7$, $C_{14}:F_7$

Order 441: $C_7^2:C_3^2$

Order 343: $C_7^3$

Order 294: $C_7^2:C_6$ x 3, $C_7:F_7$ x 3, $C_7^2:C_6$, $C_7:C_{42}$, $C_7:F_7$, $C_7:F_7$

Order 252: $C_6\times F_7$

Order 196: $C_7:D_{14}$ x 3

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

Order 126: $C_3\times F_7$, $C_{21}:C_6$

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

Order 84: $C_2\times F_7$ x 4, $C_3\times D_{14}$

Order 63: $C_{21}:C_3$

Order 49: $C_7^2$ x 3

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

Order 36: $C_6^2$

Order 28: $D_{14}$ x 3

Order 21: $C_7:C_3$ x 4, $C_{21}$

Order 18: $C_3\times C_6$ x 2

Order 14: $C_{14}$ x 3, $D_7$ x 3

Order 12: $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 4

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12348: $C_7^3:C_6^2$ ($C_1$)

Order 6174: $C_7^3:(C_3\times C_6)$ ($C_2$) x 2, $C_2\times C_7^3:C_3^2$ ($C_2$)

Order 4116: $C_2\times C_7^3:C_6$ ($C_3$) x 3, $C_2\times C_7^3:C_6$ ($C_3$)

Order 3087: $C_7^3:C_3^2$ ($C_2^2$)

Order 2058: $C_7^3:C_6$ ($C_6$) x 6, $C_7^2:C_{42}$ ($C_6$) x 3, $C_7^3:C_6$ ($C_6$) x 2, $C_7^3:C_6$ ($C_6$)

Order 1372: $C_7^2:D_{14}$ ($C_3^2$)

Order 1029: $C_7\wr C_3$ ($C_2\times C_6$) x 3, $C_7^3:C_3$ ($C_2\times C_6$)

Order 686: $C_7^3:C_2$ ($C_3\times C_6$) x 2, $C_7^2\times C_{14}$ ($C_3\times C_6$)

Order 343: $C_7^3$ ($C_6^2$)

Order 294: $C_7^2:C_6$ ($F_7$) x 3

Order 147: $C_7^2:C_3$ ($C_2\times F_7$) x 3

Order 98: $C_7\times C_{14}$ ($C_3\times F_7$) x 3

Order 49: $C_7^2$ ($C_6\times F_7$) x 3

Order 14: $C_{14}$ ($C_7:(C_3\times F_7)$) x 3

Order 7: $C_7$ ($C_7^2:C_6^2$) x 3

Order 2: $C_2$ ($C_7^3:(C_3\times C_6)$)

Order 1: $C_1$ ($C_7^3:C_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12348: $C_7^3:C_6^2$ ($C_1$)

Order 6174: $C_7^3:(C_3\times C_6)$ ($C_2$), $C_2\times C_7^3:C_3^2$ ($C_2$)

Order 4116: $C_2\times C_7^3:C_6$ ($C_3$), $C_2\times C_7^3:C_6$ ($C_3$)

Order 3087: $C_7^3:C_3^2$ ($C_2^2$)

Order 2058: $C_7^3:C_6$ ($C_6$), $C_7^3:C_6$ ($C_6$), $C_7^2:C_{42}$ ($C_6$), $C_7^3:C_6$ ($C_6$)

Order 1372: $C_7^2:D_{14}$ ($C_3^2$)

Order 1029: $C_7^3:C_3$ ($C_2\times C_6$), $C_7\wr C_3$ ($C_2\times C_6$)

Order 686: $C_7^2\times C_{14}$ ($C_3\times C_6$), $C_7^3:C_2$ ($C_3\times C_6$)

Order 343: $C_7^3$ ($C_6^2$)

Order 294: $C_7^2:C_6$ ($F_7$)

Order 147: $C_7^2:C_3$ ($C_2\times F_7$)

Order 98: $C_7\times C_{14}$ ($C_3\times F_7$)

Order 49: $C_7^2$ ($C_6\times F_7$)

Order 14: $C_{14}$ ($C_7:(C_3\times F_7)$)

Order 7: $C_7$ ($C_7^2:C_6^2$)

Order 2: $C_2$ ($C_7^3:(C_3\times C_6)$)

Order 1: $C_1$ ($C_7^3:C_6^2$)

Series

Derived series

$C_7^3:C_6^2$

$\rhd$

$C_7^3:C_6^2$

$\rhd$

$C_7^3$

$\rhd$

$C_7^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_7^3:C_6^2$

$\rhd$

$C_7^3:C_6^2$

$\rhd$

$C_2\times C_7^3:C_3^2$

$\rhd$

$C_2\times C_7^3:C_3^2$

$\rhd$

$C_7^2:C_{42}$

$\rhd$

$C_7^2:C_{42}$

$\rhd$

$C_7\wr C_3$

$\rhd$

$C_7\wr C_3$

$\rhd$

$C_7^3$

$\rhd$

$C_7^3$

$\rhd$

$C_7^2$

$\rhd$

$C_7^2$

$\rhd$

$C_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_7^3:C_6^2$

$\rhd$

$C_7^3:C_6^2$

$\rhd$

$C_7^3$

$\rhd$

$C_7^3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.