Abstract group 4032.ch: $C_3\times \SL(2,7):C_2^2$

• Label: 4032.ch
• Order: \( 2^{6} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $67$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$C_3\times \SL(2,7):C_2^2$
Order:	\(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2\times S_4\times \PGL(2,7)$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	14	21	24	28	42	84
Elements	1	127	170	48	422	48	336	1104	48	96	672	288	96	576	4032
Conjugacy classes	1	4	5	4	11	2	5	17	2	4	10	6	4	12	87
Divisions	1	4	3	4	6	1	4	10	1	1	4	3	1	3	46
Autjugacy classes	1	2	3	2	4	1	3	4	1	1	3	1	1	1	28

Dimension	1	2	3	6	7	8	12	14	16	24	32	48
Irr. complex chars.	12	0	24	12	12	18	6	0	3	0	0	0	87
Irr. rational chars.	4	4	0	8	4	4	8	4	6	1	2	1	46

Minimal presentations

Permutation degree:	$67$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	32
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $67$ $\langle(1,2,7,20,6,11,24,8)(3,12,31,43,15,34,52,21)(4,9,22,39,17,28,45,25)(5,10,23,40,19,30,47,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 5 & 6 & 0 \\ 5 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 3 & 0 \\ 1 & 4 & 0 & 4 \\ 6 & 0 & 4 & 0 \\ 0 & 1 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 6 & 3 & 3 & 1 \\ 3 & 6 & 1 & 3 \\ 0 & 6 & 5 & 4 \\ 6 & 0 & 4 & 5 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 3 & 0 \\ 3 & 4 & 0 & 4 \\ 6 & 0 & 4 & 2 \\ 0 & 1 & 3 & 3 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_3$ $\, \times\, $ $(\SL(2,7):C_2^2)$
Semidirect product:	$(\SL(2,7):C_6)$ $\,\rtimes\,$ $C_2$	$(\SL(2,7):C_2)$ $\,\rtimes\,$ $C_6$	$\SL(2,7)$ $\,\rtimes\,$ $(C_2\times C_6)$	$(C_3\times \SL(2,7))$ $\,\rtimes\,$ $C_2^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $(C_3\times \GL(3,2))$	$C_4$ . $(C_6\times \GL(3,2))$	$(C_3\times Q_8)$ . $\PSL(2,7)$	$C_{12}$ . $(C_2\times \GL(3,2))$	all 6

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 6400 subgroups in 338 conjugacy classes, 22 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2^2\times \GL(3,2)$
Commutator:	$G' \simeq$ $\SL(2,7)$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times C_6\times \GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times Q_8$	$G/\operatorname{Fit} \simeq$ $\PSL(2,7)$
Radical:	$R \simeq$ $C_3\times Q_8$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times \GL(3,2)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 4032: $C_3\times \SL(2,7):C_2^2$

Order 2016: $\SL(2,7):C_6$ x 3

Order 1344: $\SL(2,7):C_2^2$

Order 1008: $C_3\times \SL(2,7)$

Order 672: $\SL(2,7):C_2$ x 3

Order 576: $C_3\times \GL(2,3):C_2^2$ x 2

Order 504: $C_{84}.C_6$

Order 336: $\SL(2,7)$

Order 288: $\GL(2,3):C_6$ x 6, $\GL(2,3):C_6$ x 6, $Q_8.C_6^2$ x 2

Order 252: $C_{21}:C_{12}$ x 3

Order 192: $\GL(2,3):C_2^2$ x 2, $C_{12}.C_2^4$

Order 168: $C_{28}.C_6$ x 3, $Q_8\times C_{21}$

Order 144: $\SL(2,3):C_6$ x 6, $C_3\times \GL(2,3)$ x 6, $C_6.S_4$ x 2, $D_{12}:C_6$

Order 126: $C_{21}:C_6$

Order 96: $\GL(2,3):C_2$ x 6, $\GL(2,3):C_2$ x 6, $D_8:C_6$ x 6, $\SL(2,3):C_2^2$ x 4, $D_8:C_6$ x 3, $C_6\times D_8$ x 3, $C_6.C_2^4$ x 2, $\OD_{16}:C_6$

Order 84: $C_7:C_{12}$ x 9, $C_{84}$ x 3

Order 72: $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_3\times \SL(2,3)$ x 2, $Q_8\times C_3^2$

Order 64: $D_8:C_2^2$

Order 63: $C_{21}:C_3$

Order 56: $C_7\times Q_8$

Order 48: $\SL(2,3):C_2$ x 12, $C_3\times D_8$ x 9, $D_4:C_6$ x 7, $C_6\times D_4$ x 6, $\GL(2,3)$ x 6, $C_3\times \SD_{16}$ x 6, $C_3\times \OD_{16}$ x 3, $C_2\times C_{24}$ x 3, $C_2.S_4$ x 2, $D_{12}:C_2$, $C_3\times Q_{16}$

Order 42: $C_7:C_6$ x 3, $C_{42}$

Order 36: $C_3\times C_{12}$ x 3, $C_6\times S_3$ x 3, $C_3:C_{12}$

Order 32: $D_8:C_2$ x 6, $D_8:C_2$ x 3, $C_2\times D_8$ x 3, $D_4:C_2^2$ x 2, $\OD_{16}:C_2$

Order 28: $C_{28}$ x 3

Order 24: $C_3\times D_4$ x 9, $C_3\times Q_8$ x 5, $\SL(2,3)$ x 4, $C_{24}$ x 4, $C_2\times C_{12}$ x 3, $D_{12}$ x 3, $C_4\times S_3$ x 3, $C_2^2\times C_6$ x 2

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

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$

Order 16: $D_8$ x 9, $D_4:C_2$ x 7, $\SD_{16}$ x 6, $C_2\times D_4$ x 6, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $Q_{16}$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 10, $C_2\times C_6$ x 5, $D_6$ x 3, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $D_4$ x 9, $C_8$ x 4, $Q_8$ x 3, $C_2\times C_4$ x 3, $C_2^3$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 6, $S_3$ x 3

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4032: $C_3\times \SL(2,7):C_2^2$

Order 2016: $\SL(2,7):C_6$

Order 1344: $\SL(2,7):C_2^2$

Order 1008: $C_3\times \SL(2,7)$

Order 672: $\SL(2,7):C_2$

Order 576: $C_3\times \GL(2,3):C_2^2$

Order 504: $C_{84}.C_6$

Order 336: $\SL(2,7)$

Order 288: $Q_8.C_6^2$, $\GL(2,3):C_6$, $\GL(2,3):C_6$

Order 252: $C_{21}:C_{12}$

Order 192: $\GL(2,3):C_2^2$, $C_{12}.C_2^4$

Order 168: $C_{28}.C_6$ x 2, $Q_8\times C_{21}$

Order 144: $D_{12}:C_6$, $\SL(2,3):C_6$, $C_3\times \GL(2,3)$, $C_6.S_4$

Order 126: $C_{21}:C_6$

Order 96: $\SL(2,3):C_2^2$ x 2, $C_6.C_2^4$, $\GL(2,3):C_2$, $\GL(2,3):C_2$, $D_8:C_6$, $D_8:C_6$, $C_6\times D_8$, $\OD_{16}:C_6$

Order 84: $C_7:C_{12}$ x 2, $C_{84}$

Order 72: $Q_8\times C_3^2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3\times \SL(2,3)$

Order 64: $D_8:C_2^2$

Order 63: $C_{21}:C_3$

Order 56: $C_7\times Q_8$

Order 48: $D_4:C_6$ x 2, $\SL(2,3):C_2$ x 2, $C_3\times D_8$ x 2, $C_6\times D_4$, $D_{12}:C_2$, $\GL(2,3)$, $C_2.S_4$, $C_3\times Q_{16}$, $C_3\times \SD_{16}$, $C_3\times \OD_{16}$, $C_2\times C_{24}$

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

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

Order 32: $D_4:C_2^2$, $D_8:C_2$, $D_8:C_2$, $C_2\times D_8$, $\OD_{16}:C_2$

Order 28: $C_{28}$

Order 24: $C_3\times Q_8$ x 4, $\SL(2,3)$ x 2, $C_{24}$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_4\times S_3$, $C_2^2\times C_6$

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

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

Order 16: $D_8$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $C_2\times C_8$, $C_2\times D_4$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 2, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $Q_8$ x 2, $D_4$ x 2, $C_8$ x 2, $C_2^3$, $C_2\times C_4$

Order 7: $C_7$

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4032: $C_3\times \SL(2,7):C_2^2$ ($C_1$)

Order 2016: $\SL(2,7):C_6$ ($C_2$) x 3

Order 1344: $\SL(2,7):C_2^2$ ($C_3$)

Order 1008: $C_3\times \SL(2,7)$ ($C_2^2$)

Order 672: $\SL(2,7):C_2$ ($C_6$) x 3

Order 336: $\SL(2,7)$ ($C_2\times C_6$)

Order 24: $C_3\times Q_8$ ($\PSL(2,7)$)

Order 12: $C_{12}$ ($C_2\times \GL(3,2)$) x 3

Order 8: $Q_8$ ($C_3\times \GL(3,2)$)

Order 6: $C_6$ ($C_2^2\times \GL(3,2)$)

Order 4: $C_4$ ($C_6\times \GL(3,2)$) x 3

Order 3: $C_3$ ($\SL(2,7):C_2^2$)

Order 2: $C_2$ ($C_2\times C_6\times \GL(3,2)$)

Order 1: $C_1$ ($C_3\times \SL(2,7):C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4032: $C_3\times \SL(2,7):C_2^2$ ($C_1$)

Order 2016: $\SL(2,7):C_6$ ($C_2$)

Order 1344: $\SL(2,7):C_2^2$ ($C_3$)

Order 1008: $C_3\times \SL(2,7)$ ($C_2^2$)

Order 672: $\SL(2,7):C_2$ ($C_6$)

Order 336: $\SL(2,7)$ ($C_2\times C_6$)

Order 24: $C_3\times Q_8$ ($\PSL(2,7)$)

Order 12: $C_{12}$ ($C_2\times \GL(3,2)$)

Order 8: $Q_8$ ($C_3\times \GL(3,2)$)

Order 6: $C_6$ ($C_2^2\times \GL(3,2)$)

Order 4: $C_4$ ($C_6\times \GL(3,2)$)

Order 3: $C_3$ ($\SL(2,7):C_2^2$)

Order 2: $C_2$ ($C_2\times C_6\times \GL(3,2)$)

Order 1: $C_1$ ($C_3\times \SL(2,7):C_2^2$)

Series

Derived series

$C_3\times \SL(2,7):C_2^2$

$\rhd$

$\SL(2,7)$

Chief series

$C_3\times \SL(2,7):C_2^2$

$\rhd$

$C_3\times Q_8$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_3\times \SL(2,7):C_2^2$

$\rhd$

$\SL(2,7)$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times Q_8$

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 $87 \times 87$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.