Abstract group 2688.ed: $D_4\times F_8:C_6$

• Label: 2688.ed
• Order: \( 2^{7} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $14$
• Trans deg.: $56$
• Rank: $3$

Group information

Description:	$D_4\times F_8:C_6$
Order:	\(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2\wr C_2^2\times F_8:C_3$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 7, $C_3$, $C_7$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	7	12	14	28
Elements	1	95	56	32	1288	48	448	528	192	2688
Conjugacy classes	1	15	2	4	30	2	8	14	4	80
Divisions	1	15	1	4	15	1	4	7	2	50
Autjugacy classes	1	7	2	2	14	2	4	6	2	40

Dimension	1	2	3	4	6	7	12	14	28
Irr. complex chars.	24	6	16	0	4	24	0	6	0	80
Irr. rational chars.	8	10	0	2	8	8	2	10	2	50

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$56$
Rank:	$3$
Inequivalent generating triples:	$550368$

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:	${\langle a, b, c, d, e, f \mid a^{6}=b^{2}=c^{28}=d^{2}=e^{2}=f^{2}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(1,11,8)(2,3)(4,7)(5,6)(9,14,13), (1,10)(4,5)(8,11)(9,14)(12,13), (1,10,9,13,14,8,12) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $D_4$ $\, \times\, $ $(F_8:C_3)$
Semidirect product:	$(D_4\times F_8)$ $\,\rtimes\,$ $C_6$	$F_8$ $\,\rtimes\,$ $(C_6\times D_4)$	$C_2^6$ $\,\rtimes\,$ $(C_7:C_6)$	$(C_2^3\times F_8)$ $\,\rtimes\,$ $C_6$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(F_8:C_6)$ . $C_2^3$	$C_2^5$ . $(C_{14}:C_6)$	$(C_2\times F_8:C_6)$ . $C_2^2$	$C_2^2$ . $(C_2\times F_8:C_6)$	all 8
Aut. group:	$\Aut(C_2^4:C_{84})$	$\Aut(C_2^5:C_{42})$

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 13580 subgroups in 768 conjugacy classes, 76 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times F_8:C_6$
Commutator:	$G' \simeq$ $C_2\times F_8$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times F_8:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_2^4$	$G/\operatorname{Fit} \simeq$ $C_7:C_3$
Radical:	$R \simeq$ $D_4\times F_8:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_{14}:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^4$
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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Classes of subgroups up to conjugation

Order 2688: $D_4\times F_8:C_6$

Order 1344: $D_4\times F_8:C_3$ x 4, $C_2^2\times F_8:C_6$ x 2, $C_2\times F_8:C_{12}$

Order 896: $C_2^6:C_{14}$

Order 672: $C_2\times F_8:C_6$ x 9, $F_8:C_{12}$ x 2

Order 448: $D_4\times F_8$ x 4, $C_2^3\times F_8$ x 2, $C_2^4:C_{28}$

Order 384: $C_2^6:C_6$

Order 336: $F_8:C_6$ x 7, $C_2\times C_{28}:C_6$

Order 224: $C_2^2\times F_8$ x 9, $C_4\times F_8$ x 2

Order 192: $C_2^5:C_6$ x 12, $A_4\times C_2^4$ x 2, $C_2^4:C_{12}$

Order 168: $C_{28}:C_6$ x 4, $C_2\times C_{14}:C_6$ x 2, $F_8:C_3$, $C_{14}:C_{12}$

Order 128: $D_4\times C_2^4$

Order 112: $C_2\times F_8$ x 7, $D_4\times C_{14}$

Order 96: $C_2^3\times A_4$ x 21, $D_4\times A_4$ x 16, $C_2^3:C_{12}$ x 6, $C_{12}:C_2^3$

Order 84: $C_{14}:C_6$ x 9, $C_7:C_{12}$ x 2

Order 64: $D_4\times C_2^3$ x 12, $C_2^6$ x 2, $C_2^4\times C_4$

Order 56: $C_7\times D_4$ x 4, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$, $F_8$

Order 48: $C_2^2\times A_4$ x 39, $C_6\times D_4$ x 12, $C_4\times A_4$ x 4, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$

Order 42: $C_7:C_6$ x 7

Order 32: $C_2^2\times D_4$ x 38, $C_2^5$ x 21, $C_2^3\times C_4$ x 6

Order 28: $C_2\times C_{14}$ x 9, $C_{28}$ x 2

Order 24: $C_2^2\times C_6$ x 21, $C_3\times D_4$ x 16, $C_2\times A_4$ x 15, $C_2\times C_{12}$ x 6

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 67, $C_2\times D_4$ x 54, $C_2^2\times C_4$ x 10

Order 14: $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 39, $C_{12}$ x 4, $A_4$

Order 8: $C_2^3$ x 97, $D_4$ x 24, $C_2\times C_4$ x 10

Order 7: $C_7$

Order 6: $C_6$ x 15

Order 4: $C_2^2$ x 61, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2688: $D_4\times F_8:C_6$

Order 1344: $C_2^2\times F_8:C_6$, $D_4\times F_8:C_3$, $C_2\times F_8:C_{12}$

Order 896: $C_2^6:C_{14}$

Order 672: $C_2\times F_8:C_6$ x 3, $F_8:C_{12}$

Order 448: $C_2^3\times F_8$, $D_4\times F_8$, $C_2^4:C_{28}$

Order 384: $C_2^6:C_6$

Order 336: $F_8:C_6$ x 3, $C_2\times C_{28}:C_6$

Order 224: $C_2^2\times F_8$ x 3, $C_4\times F_8$

Order 192: $C_2^5:C_6$ x 5, $A_4\times C_2^4$, $C_2^4:C_{12}$

Order 168: $C_2\times C_{14}:C_6$, $F_8:C_3$, $C_{28}:C_6$, $C_{14}:C_{12}$

Order 128: $D_4\times C_2^4$

Order 112: $C_2\times F_8$ x 3, $D_4\times C_{14}$

Order 96: $C_2^3\times A_4$ x 7, $C_2^3:C_{12}$ x 4, $D_4\times A_4$ x 3, $C_{12}:C_2^3$

Order 84: $C_{14}:C_6$ x 3, $C_7:C_{12}$

Order 64: $D_4\times C_2^3$ x 5, $C_2^6$, $C_2^4\times C_4$

Order 56: $C_7\times D_4$, $C_2\times C_{28}$, $C_2^2\times C_{14}$, $F_8$

Order 48: $C_2^2\times A_4$ x 12, $C_6\times D_4$ x 5, $C_4\times A_4$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$

Order 42: $C_7:C_6$ x 3

Order 32: $C_2^2\times D_4$ x 10, $C_2^5$ x 7, $C_2^3\times C_4$ x 4

Order 28: $C_2\times C_{14}$ x 3, $C_{28}$

Order 24: $C_2^2\times C_6$ x 7, $C_2\times A_4$ x 7, $C_2\times C_{12}$ x 4, $C_3\times D_4$ x 3

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 19, $C_2\times D_4$ x 12, $C_2^2\times C_4$ x 6

Order 14: $C_{14}$ x 3

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

Order 8: $C_2^3$ x 27, $C_2\times C_4$ x 6, $D_4$ x 4

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 19, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2688: $D_4\times F_8:C_6$ ($C_1$)

Order 1344: $D_4\times F_8:C_3$ ($C_2$) x 4, $C_2^2\times F_8:C_6$ ($C_2$) x 2, $C_2\times F_8:C_{12}$ ($C_2$)

Order 896: $C_2^6:C_{14}$ ($C_3$)

Order 672: $C_2\times F_8:C_6$ ($C_2^2$) x 5, $F_8:C_{12}$ ($C_2^2$) x 2

Order 448: $D_4\times F_8$ ($C_6$) x 4, $C_2^3\times F_8$ ($C_6$) x 2, $C_2^4:C_{28}$ ($C_6$)

Order 336: $F_8:C_6$ ($D_4$) x 2, $F_8:C_6$ ($C_2^3$)

Order 224: $C_2^2\times F_8$ ($C_2\times C_6$) x 5, $C_4\times F_8$ ($C_2\times C_6$) x 2

Order 168: $F_8:C_3$ ($C_2\times D_4$)

Order 128: $D_4\times C_2^4$ ($C_7:C_3$)

Order 112: $C_2\times F_8$ ($C_3\times D_4$) x 2, $C_2\times F_8$ ($C_2^2\times C_6$)

Order 64: $D_4\times C_2^3$ ($C_7:C_6$) x 4, $C_2^6$ ($C_7:C_6$) x 2, $C_2^4\times C_4$ ($C_7:C_6$)

Order 56: $F_8$ ($C_6\times D_4$)

Order 32: $C_2^5$ ($C_{14}:C_6$) x 5, $C_2^3\times C_4$ ($C_{14}:C_6$) x 2

Order 16: $C_2^4$ ($C_{28}:C_6$) x 2, $C_2^4$ ($C_2\times C_{14}:C_6$), $C_2\times D_4$ ($F_8:C_3$)

Order 8: $D_4$ ($F_8:C_6$) x 4, $C_2^3$ ($F_8:C_6$) x 2, $C_2^3$ ($C_2\times C_{28}:C_6$), $C_2\times C_4$ ($F_8:C_6$)

Order 4: $C_2^2$ ($C_2\times F_8:C_6$) x 5, $C_4$ ($C_2\times F_8:C_6$) x 2

Order 2: $C_2$ ($D_4\times F_8:C_3$) x 2, $C_2$ ($C_2^2\times F_8:C_6$)

Order 1: $C_1$ ($D_4\times F_8:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2688: $D_4\times F_8:C_6$ ($C_1$)

Order 1344: $C_2^2\times F_8:C_6$ ($C_2$), $D_4\times F_8:C_3$ ($C_2$), $C_2\times F_8:C_{12}$ ($C_2$)

Order 896: $C_2^6:C_{14}$ ($C_3$)

Order 672: $C_2\times F_8:C_6$ ($C_2^2$) x 2, $F_8:C_{12}$ ($C_2^2$)

Order 448: $C_2^3\times F_8$ ($C_6$), $D_4\times F_8$ ($C_6$), $C_2^4:C_{28}$ ($C_6$)

Order 336: $F_8:C_6$ ($C_2^3$), $F_8:C_6$ ($D_4$)

Order 224: $C_2^2\times F_8$ ($C_2\times C_6$) x 2, $C_4\times F_8$ ($C_2\times C_6$)

Order 168: $F_8:C_3$ ($C_2\times D_4$)

Order 128: $D_4\times C_2^4$ ($C_7:C_3$)

Order 112: $C_2\times F_8$ ($C_2^2\times C_6$), $C_2\times F_8$ ($C_3\times D_4$)

Order 64: $C_2^6$ ($C_7:C_6$), $D_4\times C_2^3$ ($C_7:C_6$), $C_2^4\times C_4$ ($C_7:C_6$)

Order 56: $F_8$ ($C_6\times D_4$)

Order 32: $C_2^5$ ($C_{14}:C_6$) x 2, $C_2^3\times C_4$ ($C_{14}:C_6$)

Order 16: $C_2^4$ ($C_2\times C_{14}:C_6$), $C_2^4$ ($C_{28}:C_6$), $C_2\times D_4$ ($F_8:C_3$)

Order 8: $C_2^3$ ($F_8:C_6$), $C_2^3$ ($C_2\times C_{28}:C_6$), $D_4$ ($F_8:C_6$), $C_2\times C_4$ ($F_8:C_6$)

Order 4: $C_2^2$ ($C_2\times F_8:C_6$) x 2, $C_4$ ($C_2\times F_8:C_6$)

Order 2: $C_2$ ($C_2^2\times F_8:C_6$), $C_2$ ($D_4\times F_8:C_3$)

Order 1: $C_1$ ($D_4\times F_8:C_6$)

Series

Derived series

$D_4\times F_8:C_6$

$\rhd$

$C_2\times F_8$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$D_4\times F_8:C_6$

$\rhd$

$C_2^2\times F_8:C_6$

$\rhd$

$C_2\times F_8:C_6$

$\rhd$

$F_8:C_6$

$\rhd$

$C_2\times F_8$

$\rhd$

$F_8$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Lower central series

$D_4\times F_8:C_6$

$\rhd$

$C_2\times F_8$

$\rhd$

$F_8$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 4 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 $80 \times 80$ 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.