Abstract group 6174.bh: $C_7^3:(C_3\times C_6)$

• Label: 6174.bh
• Order: \( 2 \cdot 3^{2} \cdot 7^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{4} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $21$
• Trans deg.: $21$
• Rank: $2$

Group information

Description:	$C_7^3:(C_3\times C_6)$
Order:	\(6174\)\(\medspace = 2 \cdot 3^{2} \cdot 7^{3} \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$F_7\wr S_3$, of order \(444528\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7^{3} \)
Composition factors:	$C_2$, $C_3$ 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	21
Elements	1	343	980	2744	342	1764	6174
Conjugacy classes	1	1	8	8	21	6	45
Divisions	1	1	4	4	21	3	34
Autjugacy classes	1	1	3	3	3	1	12

Dimension	1	2	6	12	18
Irr. complex chars.	18	0	9	0	18	45
Irr. rational chars.	2	8	3	3	18	34

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$21$
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:	${\langle a, b, c, d \mid a^{6}=b^{21}=c^{7}=d^{7}=[c,d]=1, b^{a}=b^{19}c^{2}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $21$ $\langle(1,12,17,4,14,15)(2,8,21,3,11,18)(5,10,19,7,9,20)(6,13,16), (1,10,16,5,14,17,2,11,18,6,8,19,3,12,20,7,9,21,4,13,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 5 & 4 & 5 & 3 \\ 5 & 2 & 3 & 4 \\ 4 & 2 & 6 & 1 \\ 5 & 5 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 6 & 2 & 5 \\ 4 & 5 & 0 & 2 \\ 1 & 3 & 4 & 1 \\ 6 & 1 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 5 & 0 & 5 & 5 \\ 1 & 4 & 6 & 5 \\ 1 & 6 & 3 & 0 \\ 4 & 1 & 6 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 3 & 5 & 3 \\ 1 & 6 & 6 & 5 \\ 5 & 4 & 3 & 4 \\ 1 & 5 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 4 & 4 & 5 & 5 \\ 4 & 5 & 0 & 6 \\ 4 & 1 & 0 & 1 \\ 2 & 6 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 3 & 2 \\ 1 & 5 & 1 & 3 \\ 3 & 5 & 4 & 2 \\ 6 & 3 & 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Transitive group:	21T43	42T467			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_7\wr C_3)$ $\,\rtimes\,$ $C_6$	$(C_7^3:C_6)$ $\,\rtimes\,$ $C_3$	$(C_7^3:C_6)$ $\,\rtimes\,$ $C_3$	$(C_7^3:C_3)$ $\,\rtimes\,$ $C_6$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut((C_2\times C_4^3).S_4)$

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

Homology

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

Subgroups

There are 13668 subgroups in 240 conjugacy classes, 22 normal (7 characteristic).

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

Special subgroups

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

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

Order 2058: $C_7^3:C_6$ x 3, $C_7^3:C_6$

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

Order 882: $C_7:(C_3\times F_7)$ x 3

Order 686: $C_7^3:C_2$

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

Order 343: $C_7^3$

Order 294: $C_7:F_7$ x 21, $C_7:F_7$ x 6, $C_7:F_7$ x 3

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 3

Order 98: $C_7:D_7$ x 21

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

Order 49: $C_7^2$ x 21

Order 42: $F_7$ x 27, $C_3\times D_7$ x 3

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

Order 18: $C_3\times C_6$

Order 14: $D_7$ x 21

Order 9: $C_3^2$

Order 7: $C_7$ x 21

Order 6: $C_6$ x 4

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

Order 2058: $C_7^3:C_6$, $C_7^3:C_6$

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

Order 882: $C_7:(C_3\times F_7)$

Order 686: $C_7^3:C_2$

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

Order 343: $C_7^3$

Order 294: $C_7:F_7$ x 3, $C_7:F_7$, $C_7:F_7$

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

Order 126: $C_3\times F_7$

Order 98: $C_7:D_7$ x 3

Order 63: $C_{21}:C_3$

Order 49: $C_7^2$ x 3

Order 42: $F_7$ x 4, $C_3\times D_7$

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

Order 18: $C_3\times C_6$

Order 14: $D_7$ x 3

Order 9: $C_3^2$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6174: $C_7^3:(C_3\times C_6)$ ($C_1$)

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

Order 2058: $C_7^3:C_6$ ($C_3$) x 3, $C_7^3:C_6$ ($C_3$)

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

Order 686: $C_7^3:C_2$ ($C_3^2$)

Order 343: $C_7^3$ ($C_3\times C_6$)

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

Order 49: $C_7^2$ ($C_3\times F_7$) x 3

Order 7: $C_7$ ($C_7:(C_3\times F_7)$) x 3

Order 1: $C_1$ ($C_7^3:(C_3\times C_6)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6174: $C_7^3:(C_3\times C_6)$ ($C_1$)

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

Order 2058: $C_7^3:C_6$ ($C_3$), $C_7^3:C_6$ ($C_3$)

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

Order 686: $C_7^3:C_2$ ($C_3^2$)

Order 343: $C_7^3$ ($C_3\times C_6$)

Order 147: $C_7^2:C_3$ ($F_7$)

Order 49: $C_7^2$ ($C_3\times F_7$)

Order 7: $C_7$ ($C_7:(C_3\times F_7)$)

Order 1: $C_1$ ($C_7^3:(C_3\times C_6)$)

Series

Derived series

$C_7^3:(C_3\times C_6)$

$\rhd$

$C_7^3:(C_3\times C_6)$

$\rhd$

$C_7^3$

$\rhd$

$C_7^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_7^3:(C_3\times C_6)$

$\rhd$

$C_7^3:(C_3\times C_6)$

$\rhd$

$C_7^3:C_3^2$

$\rhd$

$C_7^3:C_3^2$

$\rhd$

$C_7\wr C_3$

$\rhd$

$C_7\wr C_3$

$\rhd$

$C_7^3$

$\rhd$

$C_7^3$

$\rhd$

$C_7^2$

$\rhd$

$C_7^2$

$\rhd$

$C_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_7^3:(C_3\times C_6)$

$\rhd$

$C_7^3:(C_3\times C_6)$

$\rhd$

$C_7^3$

$\rhd$

$C_7^3$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 11 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 $45 \times 45$ 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.