Abstract group 672.1093: $C_2\times D_4\times F_7$

• Label: 672.1093
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $13$
• Trans deg.: $56$
• Rank: $4$

Group information

Description:	$C_2\times D_4\times F_7$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$F_7\times C_2^5:D_4$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	7	12	14	28
Elements	1	95	14	32	322	6	112	66	24	672
Conjugacy classes	1	15	2	4	30	1	8	7	2	70
Divisions	1	15	1	4	15	1	4	7	2	50
Autjugacy classes	1	5	2	2	10	1	4	3	1	29

Dimension	1	2	4	6	12
Irr. complex chars.	48	12	0	8	2	70
Irr. rational chars.	16	20	4	8	2	50

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$56$
Rank:	$4$
Inequivalent generating quadruples:	$5745600$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{6}=d^{28}=[a,b]=[a,c]=[b,c]=[b,d]=1, d^{a}=d^{15}, d^{c}=d^{17} \rangle$
Permutation group:	Degree $13$ $\langle(2,3)(4,6)(5,7), (8,9)(10,11)(12,13), (8,10)(9,11)(12,13), (10,11)(12,13), (2,4,5)(3,6,7), (8,9)(10,11), (1,2,5,6,4,7,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 15 & 14 \\ 7 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 15 & 16 \\ 7 & 6 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/21\Z)$
Direct product:	$C_2$ $\, \times\, $ $D_4$ $\, \times\, $ $F_7$
Semidirect product:	$(D_4\times D_7)$ $\,\rtimes\,$ $C_6$	$D_7$ $\,\rtimes\,$ $(C_6\times D_4)$	$(D_4\times D_{14})$ $\,\rtimes\,$ $C_3$	$D_{14}$ $\,\rtimes\,$ $(C_3\times D_4)$	all 46
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_7:C_6)$ . $C_2^4$	$(C_2\times F_7)$ . $C_2^3$	$C_2$ . $(C_2^3\times F_7)$	$D_{14}$ . $(C_2^2\times C_6)$	all 10
Aut. group:	$\Aut(C_{56}:C_2)$	$\Aut(C_{14}:C_8)$	$\Aut(C_7:\OD_{16})$	$\Aut(C_7:D_8)$	all 26

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

Homology

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

Subgroups

There are 2318 subgroups in 472 conjugacy classes, 175 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times F_7$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{14}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_2\times D_4\times F_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 672: $C_2\times D_4\times F_7$

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

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

Order 168: $C_2^2\times F_7$ x 19, $D_{14}:C_6$ 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, $C_{14}:C_{12}$, $C_{14}:C_{12}$

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: $C_{12}:C_2^3$

Order 84: $C_2\times F_7$ x 30, $C_{14}:C_6$ x 9, $C_7:C_{12}$ x 2, $C_7:C_{12}$ x 2

Order 56: $C_2\times D_{14}$ x 19, $C_7:D_4$ 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, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$

Order 42: $F_7$ x 8, $C_7:C_6$ x 7

Order 32: $C_2^2\times D_4$

Order 28: $D_{14}$ x 30, $C_2\times C_{14}$ x 9, $C_{28}$ x 2, $C_7:C_4$ x 2

Order 24: $C_2^2\times C_6$ x 21, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 6

Order 21: $C_7:C_3$

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 8, $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 39, $C_{12}$ x 4

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 7: $C_7$

Order 6: $C_6$ x 15

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $C_2\times D_4\times F_7$

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

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

Order 168: $C_2^2\times F_7$ x 4, $C_{28}:C_6$, $C_4\times F_7$, $C_2\times C_{14}:C_6$, $C_{28}:C_6$, $C_{14}:C_{12}$, $D_{14}:C_6$, $C_{14}:C_{12}$

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

Order 96: $C_{12}:C_2^3$

Order 84: $C_2\times F_7$ x 5, $C_{14}:C_6$ x 3, $C_7:C_{12}$, $C_7:C_{12}$

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

Order 48: $C_6\times D_4$ x 4, $C_2^3\times C_6$, $C_2^2\times C_{12}$

Order 42: $C_7:C_6$ x 3, $F_7$ x 2

Order 32: $C_2^2\times D_4$

Order 28: $D_{14}$ x 5, $C_2\times C_{14}$ x 3, $C_{28}$, $C_7:C_4$

Order 24: $C_2^2\times C_6$ x 5, $C_2\times C_{12}$ x 3, $C_3\times D_4$ x 3

Order 21: $C_7:C_3$

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

Order 14: $C_{14}$ x 3, $D_7$ x 2

Order 12: $C_2\times C_6$ x 8, $C_{12}$ x 2

Order 8: $C_2^3$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3

Order 7: $C_7$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 8, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $C_2\times D_4\times F_7$ ($C_1$)

Order 336: $D_4\times F_7$ ($C_2$) x 8, $C_2^3\times F_7$ ($C_2$) x 2, $C_2^3:F_7$ ($C_2$) x 2, $C_2\times C_{28}:C_6$ ($C_2$), $D_{28}:C_6$ ($C_2$), $D_{14}:C_{12}$ ($C_2$)

Order 224: $D_4\times D_{14}$ ($C_3$)

Order 168: $C_2^2\times F_7$ ($C_2^2$) x 11, $D_{14}:C_6$ ($C_2^2$) x 8, $C_{28}:C_6$ ($C_2^2$) x 4, $C_4\times F_7$ ($C_2^2$) x 4, $C_{28}:C_6$ ($C_2^2$) x 4, $C_2\times C_{14}:C_6$ ($C_2^2$) x 2, $C_{14}:C_{12}$ ($C_2^2$), $C_{14}:C_{12}$ ($C_2^2$)

Order 112: $D_4\times D_7$ ($C_6$) x 8, $C_2^2\times D_{14}$ ($C_6$) x 2, $C_{14}:D_4$ ($C_6$) x 2, $D_4\times C_{14}$ ($C_6$), $C_2\times D_{28}$ ($C_6$), $C_4\times D_{14}$ ($C_6$)

Order 84: $C_2\times F_7$ ($C_2^3$) x 6, $C_{14}:C_6$ ($C_2^3$) x 5, $C_2\times F_7$ ($D_4$) x 4, $C_7:C_{12}$ ($C_2^3$) x 2, $C_7:C_{12}$ ($C_2^3$) x 2

Order 56: $C_2\times D_{14}$ ($C_2\times C_6$) x 11, $C_7:D_4$ ($C_2\times C_6$) x 8, $C_7\times D_4$ ($C_2\times C_6$) x 4, $D_{28}$ ($C_2\times C_6$) x 4, $C_4\times D_7$ ($C_2\times C_6$) x 4, $C_2^2\times C_{14}$ ($C_2\times C_6$) x 2, $C_2\times C_{28}$ ($C_2\times C_6$), $C_{14}:C_4$ ($C_2\times C_6$)

Order 42: $F_7$ ($C_2\times D_4$) x 4, $C_7:C_6$ ($C_2\times D_4$) x 2, $C_7:C_6$ ($C_2^4$)

Order 28: $D_{14}$ ($C_2^2\times C_6$) x 6, $C_2\times C_{14}$ ($C_2^2\times C_6$) x 5, $D_{14}$ ($C_3\times D_4$) x 4, $C_{28}$ ($C_2^2\times C_6$) x 2, $C_7:C_4$ ($C_2^2\times C_6$) x 2

Order 21: $C_7:C_3$ ($C_2^2\times D_4$)

Order 16: $C_2\times D_4$ ($F_7$)

Order 14: $D_7$ ($C_6\times D_4$) x 4, $C_{14}$ ($C_6\times D_4$) x 2, $C_{14}$ ($C_2^3\times C_6$)

Order 8: $D_4$ ($C_2\times F_7$) x 4, $C_2^3$ ($C_2\times F_7$) x 2, $C_2\times C_4$ ($C_2\times F_7$)

Order 7: $C_7$ ($C_{12}:C_2^3$)

Order 4: $C_2^2$ ($C_2^2\times F_7$) x 5, $C_4$ ($C_2^2\times F_7$) x 2

Order 2: $C_2$ ($D_4\times F_7$) x 2, $C_2$ ($C_2^3\times F_7$)

Order 1: $C_1$ ($C_2\times D_4\times F_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $C_2\times D_4\times F_7$ ($C_1$)

Order 336: $C_2^3\times F_7$ ($C_2$), $C_2\times C_{28}:C_6$ ($C_2$), $C_2^3:F_7$ ($C_2$), $D_4\times F_7$ ($C_2$), $D_{28}:C_6$ ($C_2$), $D_{14}:C_{12}$ ($C_2$)

Order 224: $D_4\times D_{14}$ ($C_3$)

Order 168: $C_2^2\times F_7$ ($C_2^2$) x 3, $C_{28}:C_6$ ($C_2^2$), $C_4\times F_7$ ($C_2^2$), $C_2\times C_{14}:C_6$ ($C_2^2$), $C_{28}:C_6$ ($C_2^2$), $C_{14}:C_{12}$ ($C_2^2$), $D_{14}:C_6$ ($C_2^2$), $C_{14}:C_{12}$ ($C_2^2$)

Order 112: $C_2^2\times D_{14}$ ($C_6$), $D_4\times C_{14}$ ($C_6$), $C_{14}:D_4$ ($C_6$), $D_4\times D_7$ ($C_6$), $C_2\times D_{28}$ ($C_6$), $C_4\times D_{14}$ ($C_6$)

Order 84: $C_{14}:C_6$ ($C_2^3$) x 2, $C_2\times F_7$ ($C_2^3$) x 2, $C_2\times F_7$ ($D_4$), $C_7:C_{12}$ ($C_2^3$), $C_7:C_{12}$ ($C_2^3$)

Order 56: $C_2\times D_{14}$ ($C_2\times C_6$) x 3, $C_7\times D_4$ ($C_2\times C_6$), $C_2\times C_{28}$ ($C_2\times C_6$), $C_7:D_4$ ($C_2\times C_6$), $C_{14}:C_4$ ($C_2\times C_6$), $D_{28}$ ($C_2\times C_6$), $C_4\times D_7$ ($C_2\times C_6$), $C_2^2\times C_{14}$ ($C_2\times C_6$)

Order 42: $C_7:C_6$ ($C_2^4$), $C_7:C_6$ ($C_2\times D_4$), $F_7$ ($C_2\times D_4$)

Order 28: $C_2\times C_{14}$ ($C_2^2\times C_6$) x 2, $D_{14}$ ($C_2^2\times C_6$) x 2, $D_{14}$ ($C_3\times D_4$), $C_{28}$ ($C_2^2\times C_6$), $C_7:C_4$ ($C_2^2\times C_6$)

Order 21: $C_7:C_3$ ($C_2^2\times D_4$)

Order 16: $C_2\times D_4$ ($F_7$)

Order 14: $C_{14}$ ($C_2^3\times C_6$), $C_{14}$ ($C_6\times D_4$), $D_7$ ($C_6\times D_4$)

Order 8: $C_2^3$ ($C_2\times F_7$), $D_4$ ($C_2\times F_7$), $C_2\times C_4$ ($C_2\times F_7$)

Order 7: $C_7$ ($C_{12}:C_2^3$)

Order 4: $C_2^2$ ($C_2^2\times F_7$) x 2, $C_4$ ($C_2^2\times F_7$)

Order 2: $C_2$ ($C_2^3\times F_7$), $C_2$ ($D_4\times F_7$)

Order 1: $C_1$ ($C_2\times D_4\times F_7$)

Series

Derived series

$C_2\times D_4\times F_7$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_2\times D_4\times F_7$

$\rhd$

$C_2\times C_{28}:C_6$

$\rhd$

$C_{28}:C_6$

$\rhd$

$C_7:C_{12}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_2\times D_4\times F_7$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 50 larger groups in the database.

This group is a maximal quotient of 109 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.