Abstract group 12348.u: $C_3\times \He_7:D_6$

• Label: 12348.u
• Order: \( 2^{2} \cdot 3^{2} \cdot 7^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $101$
• Trans deg.: $294$
• Rank: $2$

Group information

Description:	$C_3\times \He_7:D_6$
Order:	\(12348\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{3} \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$\He_7:(C_6^2:C_2^2)$, of order \(49392\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{3} \)
Composition factors:	$C_2$ x 2, $C_3$ x 2, $C_7$ x 3
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	343	296	980	342	2058	2448	5880	12348
Conjugacy classes	1	3	5	9	10	9	38	36	111
Divisions	1	3	3	5	4	3	6	5	30
Autjugacy classes	1	2	3	4	3	2	5	4	24

Dimension	1	2	4	6	12	14	18	24	36	42	84	168
Irr. complex chars.	12	6	0	36	3	54	0	0	0	0	0	0	111
Irr. rational chars.	4	6	2	0	1	0	4	1	4	2	4	2	30

Minimal presentations

Permutation degree:	$101$
Transitive degree:	$294$
Rank:	$2$
Inequivalent generating pairs:	$672$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	28	84
Arbitrary	14	16	44

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{6}=c^{7}=d^{7}=e^{7}=[b,e]=[c,e]=[d,e]=1, a^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $101$ $\langle(2,7)(4,17)(5,22)(6,9)(8,32)(10,14)(11,15)(16,37)(18,42)(19,55)(20,54)(21,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 & 6 & 0 & 0 \\ 2 & 0 & 6 & 0 \\ 0 & 5 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 5 & 0 & 5 & 6 \\ 6 & 5 & 0 & 1 \\ 6 & 5 & 3 & 1 \\ 5 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 4 & 2 & 3 \\ 5 & 6 & 1 & 2 \\ 2 & 4 & 5 & 1 \\ 6 & 2 & 3 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 1 & 3 \\ 3 & 6 & 5 & 1 \\ 3 & 5 & 3 & 6 \\ 6 & 3 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 1 & 6 & 4 & 4 \\ 4 & 5 & 3 & 4 \\ 6 & 2 & 4 & 1 \\ 2 & 6 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 3 & 2 \\ 1 & 5 & 1 & 3 \\ 3 & 5 & 4 & 2 \\ 6 & 3 & 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_3$ $\, \times\, $ $(\He_7:D_6)$
Semidirect product:	$(\He_7:C_6)$ $\,\rtimes\,$ $S_3$	$(\He_7:S_3)$ $\,\rtimes\,$ $C_6$ (2)	$\He_7$ $\,\rtimes\,$ $(C_6\times S_3)$	$(C_3\times \He_7)$ $\,\rtimes\,$ $D_6$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{21}$ . $(C_7^2:D_6)$	$C_7$ . $(C_3\times C_7^2:D_6)$			more information

Elements of the group are displayed as matrices in $\GL_{4}(\F_{7})$.

Homology

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

Subgroups

There are 8916 subgroups in 130 conjugacy classes, 18 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $\He_7:D_6$
Commutator:	$G' \simeq$ $\He_7:C_3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_7$	$G/\Phi \simeq$ $C_3\times C_7^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times \He_7$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_3\times \He_7:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{21}$	$G/\operatorname{soc} \simeq$ $C_7^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $\He_7$

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 12348: $C_3\times \He_7:D_6$

Order 6174: $C_3\times \He_7:S_3$ x 2, $C_3\times C_7^2:F_7$

Order 4116: $\He_7:D_6$, $C_{21}.D_7^2$

Order 3087: $\He_7:C_3^2$

Order 2058: $\He_7:S_3$ x 2, $C_7^2:C_{42}$ x 2, $C_7^2:F_7$ x 2, $\He_7:C_6$

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

Order 1029: $\He_7:C_3$ x 2, $C_3\times \He_7$

Order 882: $C_{21}\times F_7$

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

Order 588: $C_3\times D_7^2$ x 2

Order 441: $C_{21}:C_{21}$

Order 343: $\He_7$

Order 294: $D_7\times C_{21}$ x 5, $C_7\times F_7$ x 3, $C_{21}:D_7$ x 2

Order 252: $C_3\times D_{42}$

Order 196: $D_7^2$ x 2

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

Order 126: $C_3\times D_{21}$ x 2, $C_3\times F_7$, $C_3\times C_{42}$

Order 98: $C_7\times D_7$ x 5, $C_7:D_7$ x 2

Order 84: $C_3\times D_{14}$ x 3, $D_{42}$

Order 63: $C_3\times C_{21}$, $C_{21}:C_3$

Order 49: $C_7^2$ x 3

Order 42: $C_3\times D_7$ x 7, $C_{42}$ x 5, $F_7$ x 3, $D_{21}$ x 2

Order 36: $C_6\times S_3$

Order 28: $D_{14}$ x 3

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

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$

Order 14: $D_7$ x 7, $C_{14}$ x 3

Order 12: $C_2\times C_6$, $D_6$

Order 9: $C_3^2$

Order 7: $C_7$ x 4

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12348: $C_3\times \He_7:D_6$

Order 6174: $C_3\times C_7^2:F_7$, $C_3\times \He_7:S_3$

Order 4116: $\He_7:D_6$, $C_{21}.D_7^2$

Order 3087: $\He_7:C_3^2$

Order 2058: $C_7^2:F_7$ x 2, $\He_7:C_6$, $\He_7:S_3$, $C_7^2:C_{42}$

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

Order 1029: $\He_7:C_3$ x 2, $C_3\times \He_7$

Order 882: $C_{21}\times F_7$

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

Order 588: $C_3\times D_7^2$

Order 441: $C_{21}:C_{21}$

Order 343: $\He_7$

Order 294: $D_7\times C_{21}$ x 3, $C_7\times F_7$ x 2, $C_{21}:D_7$

Order 252: $C_3\times D_{42}$

Order 196: $D_7^2$

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

Order 126: $C_3\times F_7$, $C_3\times C_{42}$, $C_3\times D_{21}$

Order 98: $C_7\times D_7$ x 3, $C_7:D_7$

Order 84: $C_3\times D_{14}$ x 2, $D_{42}$

Order 63: $C_3\times C_{21}$, $C_{21}:C_3$

Order 49: $C_7^2$ x 2

Order 42: $C_{42}$ x 4, $C_3\times D_7$ x 4, $F_7$ x 2, $D_{21}$

Order 36: $C_6\times S_3$

Order 28: $D_{14}$ x 2

Order 21: $C_{21}$ x 5, $C_7:C_3$ x 2

Order 18: $C_3\times C_6$, $C_3\times S_3$

Order 14: $D_7$ x 4, $C_{14}$ x 2

Order 12: $C_2\times C_6$, $D_6$

Order 9: $C_3^2$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 4, $S_3$

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12348: $C_3\times \He_7:D_6$ ($C_1$)

Order 6174: $C_3\times \He_7:S_3$ ($C_2$) x 2, $C_3\times C_7^2:F_7$ ($C_2$)

Order 4116: $\He_7:D_6$ ($C_3$)

Order 3087: $\He_7:C_3^2$ ($C_2^2$)

Order 2058: $\He_7:S_3$ ($C_6$) x 2, $\He_7:C_6$ ($S_3$), $C_7^2:F_7$ ($C_6$)

Order 1029: $C_3\times \He_7$ ($D_6$), $\He_7:C_3$ ($C_2\times C_6$)

Order 686: $C_7^2:D_7$ ($C_3\times S_3$)

Order 343: $\He_7$ ($C_6\times S_3$)

Order 21: $C_{21}$ ($C_7^2:D_6$)

Order 7: $C_7$ ($C_3\times C_7^2:D_6$)

Order 3: $C_3$ ($\He_7:D_6$)

Order 1: $C_1$ ($C_3\times \He_7:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12348: $C_3\times \He_7:D_6$ ($C_1$)

Order 6174: $C_3\times C_7^2:F_7$ ($C_2$), $C_3\times \He_7:S_3$ ($C_2$)

Order 4116: $\He_7:D_6$ ($C_3$)

Order 3087: $\He_7:C_3^2$ ($C_2^2$)

Order 2058: $\He_7:C_6$ ($S_3$), $\He_7:S_3$ ($C_6$), $C_7^2:F_7$ ($C_6$)

Order 1029: $C_3\times \He_7$ ($D_6$), $\He_7:C_3$ ($C_2\times C_6$)

Order 686: $C_7^2:D_7$ ($C_3\times S_3$)

Order 343: $\He_7$ ($C_6\times S_3$)

Order 21: $C_{21}$ ($C_7^2:D_6$)

Order 7: $C_7$ ($C_3\times C_7^2:D_6$)

Order 3: $C_3$ ($\He_7:D_6$)

Order 1: $C_1$ ($C_3\times \He_7:D_6$)

Series

Derived series

$C_3\times \He_7:D_6$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

Chief series

$C_3\times \He_7:D_6$

$\rhd$

$C_3\times \He_7:S_3$

$\rhd$

$\He_7:C_3^2$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_3\times \He_7:D_6$

$\rhd$

$\He_7:C_3$

Upper central series

$C_1$

$\lhd$

$C_3$

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 $111 \times 111$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.