Abstract group 2016.dc: $C_2\times S_4\times F_7$

• Label: 2016.dc
• Order: \( 2^{5} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $13$
• Trans deg.: $42$
• Rank: $3$

Group information

Description:	$C_2\times S_4\times F_7$
Order:	\(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^2\times S_4\times F_7$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $C_7$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42
Elements	1	159	134	96	1002	6	336	114	48	72	48	2016
Conjugacy classes	1	11	5	4	31	1	8	5	1	2	1	70
Divisions	1	11	3	4	17	1	4	5	1	2	1	50
Autjugacy classes	1	7	5	2	20	1	4	4	1	1	1	47

Dimension	1	2	3	4	6	12	18
Irr. complex chars.	24	12	24	0	4	2	4	70
Irr. rational chars.	8	12	8	4	12	2	4	50

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$42$
Rank:	$3$
Inequivalent generating triples:	$524160$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	9	9	9

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{6}=c^{2}=d^{14}=e^{2}=[a,e]=[b,e]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(2,3)(4,6)(8,10,12,11,13,9), (1,2)(3,5)(4,6)(7,8,10)(9,11,12), (1,3,6,5,2,4)(7,9,8)(10,13,12)\rangle$
Transitive group:	42T260				more information
Direct product:	$C_2$ $\, \times\, $ $S_4$ $\, \times\, $ $F_7$
Semidirect product:	$(D_7\times S_4)$ $\,\rtimes\,$ $C_6$ (4)	$D_7$ $\,\rtimes\,$ $(C_6\times S_4)$ (2)	$(S_4\times D_{14})$ $\,\rtimes\,$ $C_3$	$D_{14}$ $\,\rtimes\,$ $(C_3\times S_4)$	all 36
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(Q_8\times D_7)$	$\Aut(D_{28}:C_2)$	$\Aut(C_6\times D_{14})$	$\Aut(C_{14}.S_4)$	all 52

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 6532 subgroups in 544 conjugacy classes, 61 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_4\times F_7$
Commutator:	$G' \simeq$ $C_7\times A_4$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times S_4\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{14}$	$G/\operatorname{Fit} \simeq$ $C_6\times S_3$
Radical:	$R \simeq$ $C_2\times S_4\times F_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_6\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
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

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 2016: $C_2\times S_4\times F_7$

Order 1008: $S_4\times F_7$ x 4, $C_2\times A_4\times F_7$, $C_2\times A_4:F_7$, $C_7:C_6\times S_4$

Order 672: $S_4\times D_{14}$, $C_2\times D_4\times F_7$

Order 504: $A_4\times F_7$ x 2, $A_4:F_7$ x 2, $C_7:C_3\times S_4$ x 2, $C_7:C_6\times A_4$, $D_6\times F_7$

Order 336: $D_4\times F_7$ x 8, $D_7\times S_4$ x 4, $C_2^3\times F_7$ x 2, $C_2^3:F_7$ x 2, $D_{14}:A_4$, $A_4\times D_{14}$, $C_{14}:S_4$, $C_{14}\times S_4$, $C_2\times C_{28}:C_6$, $D_{28}:C_6$, $D_{14}:C_{12}$

Order 288: $C_2\times C_6\times S_4$

Order 252: $S_3\times F_7$ x 4, $C_{42}:C_6$, $C_{42}:C_6$, $C_6\times F_7$, $A_4\times C_7:C_3$

Order 224: $D_4\times D_{14}$

Order 168: $C_2^2\times F_7$ x 15, $D_{14}:C_6$ x 8, $C_{28}:C_6$ x 4, $C_4\times F_7$ x 4, $C_{28}:C_6$ x 4, $C_2\times C_{14}:C_6$ x 2, $D_7:A_4$ x 2, $A_4\times D_7$ x 2, $C_7:S_4$ x 2, $C_7\times S_4$ x 2, $C_{14}:A_4$, $A_4\times C_{14}$, $S_3\times D_{14}$, $C_{14}:C_{12}$, $C_{14}:C_{12}$

Order 144: $C_6\times S_4$ x 6, $C_2^2:C_6^2$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$ x 2, $C_3\times F_7$ x 2, $C_{21}:C_6$

Order 112: $D_4\times D_7$ x 8, $C_2^2\times D_{14}$ x 2, $C_{14}:D_4$ x 2, $D_4\times C_{14}$, $C_2\times D_{28}$, $C_4\times D_{14}$

Order 96: $C_2^2\times S_4$, $C_{12}:C_2^3$

Order 84: $C_2\times F_7$ x 22, $C_{14}:C_6$ x 7, $S_3\times D_7$ x 4, $C_7:C_{12}$ x 2, $C_7:C_{12}$ x 2, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$, $C_7:A_4$, $C_7\times A_4$

Order 72: $C_3\times S_4$ x 4, $C_6\times A_4$ x 3, $C_6\times D_6$

Order 63: $C_{21}:C_3$

Order 56: $C_2\times D_{14}$ x 15, $C_7:D_4$ x 8, $C_7\times D_4$ x 4, $D_{28}$ x 4, $C_4\times D_7$ x 4, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$

Order 48: $C_6\times D_4$ x 12, $C_2\times S_4$ x 6, $C_2^3\times C_6$ x 2, $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$

Order 42: $F_7$ x 8, $C_7:C_6$ x 6, $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2, $C_{42}$

Order 36: $C_6\times S_3$ x 6, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

Order 28: $D_{14}$ x 21, $C_2\times C_{14}$ x 7, $C_{28}$ x 2, $C_7:C_4$ x 2

Order 24: $C_2^2\times C_6$ x 17, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 6, $C_2\times A_4$ x 6, $S_4$ x 4, $C_2\times D_6$

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

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 14: $D_7$ x 6, $C_{14}$ x 5

Order 12: $C_2\times C_6$ x 30, $D_6$ x 6, $C_{12}$ x 4, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

Order 7: $C_7$

Order 6: $C_6$ x 17, $S_3$ x 4

Order 4: $C_2^2$ x 28, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2016: $C_2\times S_4\times F_7$

Order 1008: $C_2\times A_4\times F_7$, $C_2\times A_4:F_7$, $C_7:C_6\times S_4$, $S_4\times F_7$

Order 672: $S_4\times D_{14}$, $C_2\times D_4\times F_7$

Order 504: $C_7:C_6\times A_4$, $D_6\times F_7$, $A_4\times F_7$, $A_4:F_7$, $C_7:C_3\times S_4$

Order 336: $C_2^3\times F_7$ x 2, $C_2^3:F_7$ x 2, $D_4\times F_7$ x 2, $D_{14}:A_4$, $A_4\times D_{14}$, $C_{14}:S_4$, $C_{14}\times S_4$, $D_7\times S_4$, $C_2\times C_{28}:C_6$, $D_{28}:C_6$, $D_{14}:C_{12}$

Order 288: $C_2\times C_6\times S_4$

Order 252: $C_{42}:C_6$, $C_{42}:C_6$, $C_6\times F_7$, $A_4\times C_7:C_3$, $S_3\times F_7$

Order 224: $D_4\times D_{14}$

Order 168: $C_2^2\times F_7$ x 9, $D_{14}:C_6$ x 3, $C_2\times C_{14}:C_6$ x 2, $C_{28}:C_6$ x 2, $C_{28}:C_6$, $C_4\times F_7$, $C_{14}:A_4$, $A_4\times C_{14}$, $S_3\times D_{14}$, $D_7:A_4$, $A_4\times D_7$, $C_7:S_4$, $C_7\times S_4$, $C_{14}:C_{12}$, $C_{14}:C_{12}$

Order 144: $C_6\times S_4$ x 3, $C_2^2:C_6^2$

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

Order 112: $C_2^2\times D_{14}$ x 2, $C_{14}:D_4$ x 2, $D_4\times D_7$ x 2, $D_4\times C_{14}$, $C_2\times D_{28}$, $C_4\times D_{14}$

Order 96: $C_2^2\times S_4$, $C_{12}:C_2^3$

Order 84: $C_2\times F_7$ x 12, $C_{14}:C_6$ x 6, $S_3\times D_7$, $C_7:C_{12}$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$, $C_7:A_4$, $C_7\times A_4$, $C_7:C_{12}$

Order 72: $C_6\times A_4$ x 2, $C_3\times S_4$ x 2, $C_6\times D_6$

Order 63: $C_{21}:C_3$

Order 56: $C_2\times D_{14}$ x 9, $C_7:D_4$ x 3, $C_7\times D_4$ x 2, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$

Order 48: $C_6\times D_4$ x 6, $C_2\times S_4$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$

Order 42: $C_7:C_6$ x 5, $F_7$ x 4, $C_{42}$, $D_{21}$, $C_3\times D_7$, $S_3\times C_7$

Order 36: $C_6\times S_3$ x 3, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

Order 28: $D_{14}$ x 11, $C_2\times C_{14}$ x 6, $C_{28}$, $C_7:C_4$

Order 24: $C_2^2\times C_6$ x 11, $C_3\times D_4$ x 6, $C_2\times A_4$ x 4, $C_2\times C_{12}$ x 3, $S_4$ x 2, $C_2\times D_6$

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

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

Order 16: $C_2\times D_4$ x 6, $C_2^4$ x 2, $C_2^2\times C_4$

Order 14: $C_{14}$ x 4, $D_7$ x 3

Order 12: $C_2\times C_6$ x 19, $D_6$ x 3, $A_4$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 11, $D_4$ x 6, $C_2\times C_4$ x 3

Order 7: $C_7$

Order 6: $C_6$ x 11, $S_3$ x 2

Order 4: $C_2^2$ x 17, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2016: $C_2\times S_4\times F_7$ ($C_1$)

Order 1008: $S_4\times F_7$ ($C_2$) x 4, $C_2\times A_4\times F_7$ ($C_2$), $C_2\times A_4:F_7$ ($C_2$), $C_7:C_6\times S_4$ ($C_2$)

Order 672: $S_4\times D_{14}$ ($C_3$)

Order 504: $A_4\times F_7$ ($C_2^2$) x 2, $A_4:F_7$ ($C_2^2$) x 2, $C_7:C_3\times S_4$ ($C_2^2$) x 2, $C_7:C_6\times A_4$ ($C_2^2$)

Order 336: $D_7\times S_4$ ($C_6$) x 4, $A_4\times D_{14}$ ($C_6$), $C_2^3\times F_7$ ($S_3$), $C_{14}:S_4$ ($C_6$), $C_{14}\times S_4$ ($C_6$)

Order 252: $A_4\times C_7:C_3$ ($C_2^3$)

Order 168: $A_4\times D_7$ ($C_2\times C_6$) x 2, $C_2^2\times F_7$ ($D_6$) x 2, $C_7:S_4$ ($C_2\times C_6$) x 2, $C_7\times S_4$ ($C_2\times C_6$) x 2, $A_4\times C_{14}$ ($C_2\times C_6$), $C_2\times C_{14}:C_6$ ($D_6$)

Order 112: $C_2^2\times D_{14}$ ($C_3\times S_3$)

Order 84: $C_{14}:C_6$ ($C_2\times D_6$), $C_2\times F_7$ ($S_4$), $C_7\times A_4$ ($C_2^2\times C_6$)

Order 56: $C_2\times D_{14}$ ($C_6\times S_3$) x 2, $C_2^2\times C_{14}$ ($C_6\times S_3$)

Order 48: $C_2\times S_4$ ($F_7$)

Order 42: $F_7$ ($C_2\times S_4$) x 2, $C_7:C_6$ ($C_2\times S_4$)

Order 28: $C_2\times C_{14}$ ($C_6\times D_6$), $D_{14}$ ($C_3\times S_4$)

Order 24: $S_4$ ($C_2\times F_7$) x 2, $C_2\times A_4$ ($C_2\times F_7$)

Order 21: $C_7:C_3$ ($C_2^2\times S_4$)

Order 14: $D_7$ ($C_6\times S_4$) x 2, $C_{14}$ ($C_6\times S_4$)

Order 12: $A_4$ ($C_2^2\times F_7$)

Order 8: $C_2^3$ ($S_3\times F_7$)

Order 7: $C_7$ ($C_2\times C_6\times S_4$)

Order 4: $C_2^2$ ($D_6\times F_7$)

Order 2: $C_2$ ($S_4\times F_7$)

Order 1: $C_1$ ($C_2\times S_4\times F_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2016: $C_2\times S_4\times F_7$ ($C_1$)

Order 1008: $C_2\times A_4\times F_7$ ($C_2$), $C_2\times A_4:F_7$ ($C_2$), $C_7:C_6\times S_4$ ($C_2$), $S_4\times F_7$ ($C_2$)

Order 672: $S_4\times D_{14}$ ($C_3$)

Order 504: $C_7:C_6\times A_4$ ($C_2^2$), $A_4\times F_7$ ($C_2^2$), $A_4:F_7$ ($C_2^2$), $C_7:C_3\times S_4$ ($C_2^2$)

Order 336: $A_4\times D_{14}$ ($C_6$), $C_2^3\times F_7$ ($S_3$), $C_{14}:S_4$ ($C_6$), $C_{14}\times S_4$ ($C_6$), $D_7\times S_4$ ($C_6$)

Order 252: $A_4\times C_7:C_3$ ($C_2^3$)

Order 168: $A_4\times C_{14}$ ($C_2\times C_6$), $C_2\times C_{14}:C_6$ ($D_6$), $A_4\times D_7$ ($C_2\times C_6$), $C_2^2\times F_7$ ($D_6$), $C_7:S_4$ ($C_2\times C_6$), $C_7\times S_4$ ($C_2\times C_6$)

Order 112: $C_2^2\times D_{14}$ ($C_3\times S_3$)

Order 84: $C_{14}:C_6$ ($C_2\times D_6$), $C_2\times F_7$ ($S_4$), $C_7\times A_4$ ($C_2^2\times C_6$)

Order 56: $C_2^2\times C_{14}$ ($C_6\times S_3$), $C_2\times D_{14}$ ($C_6\times S_3$)

Order 48: $C_2\times S_4$ ($F_7$)

Order 42: $C_7:C_6$ ($C_2\times S_4$), $F_7$ ($C_2\times S_4$)

Order 28: $C_2\times C_{14}$ ($C_6\times D_6$), $D_{14}$ ($C_3\times S_4$)

Order 24: $C_2\times A_4$ ($C_2\times F_7$), $S_4$ ($C_2\times F_7$)

Order 21: $C_7:C_3$ ($C_2^2\times S_4$)

Order 14: $C_{14}$ ($C_6\times S_4$), $D_7$ ($C_6\times S_4$)

Order 12: $A_4$ ($C_2^2\times F_7$)

Order 8: $C_2^3$ ($S_3\times F_7$)

Order 7: $C_7$ ($C_2\times C_6\times S_4$)

Order 4: $C_2^2$ ($D_6\times F_7$)

Order 2: $C_2$ ($S_4\times F_7$)

Order 1: $C_1$ ($C_2\times S_4\times F_7$)

Series

Derived series

$C_2\times S_4\times F_7$

$\rhd$

$C_7\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2\times S_4\times F_7$

$\rhd$

$C_7:C_6\times S_4$

$\rhd$

$C_7:C_6\times A_4$

$\rhd$

$A_4\times C_7:C_3$

$\rhd$

$C_7\times A_4$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2\times S_4\times F_7$

$\rhd$

$C_7\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 23 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $70 \times 70$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $50 \times 50$ rational character table.