Abstract group 448.1362: $C_2^4:C_{28}$

• Label: 448.1362
• Order: \( 2^{6} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $56$
• Rank: $2$

Group information

Description:	$C_2^4:C_{28}$
Order:	\(448\)\(\medspace = 2^{6} \cdot 7 \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$D_4\times F_8:C_3$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_7$
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	4	7	14	28
Elements	1	31	32	48	144	192	448
Conjugacy classes	1	7	8	6	18	24	64
Divisions	1	7	4	1	3	2	18
Autjugacy classes	1	5	2	2	4	2	16

Dimension	1	2	6	7	12	14
Irr. complex chars.	56	0	0	8	0	0	64
Irr. rational chars.	4	2	4	4	2	2	18

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$56$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	8	9	9

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{28}=b^{2}=c^{2}=d^{2}=e^{2}=[a,b]=[b,c]=[b,d]=[b,e]=[c,d]=[c,e]=[d,e]=1, c^{a}=cde, d^{a}=ce, e^{a}=c \rangle$
Permutation group:	Degree $14$ $\langle(13,14), (9,10,11,12)(13,14), (2,3,4,6,8,5,7), (9,11)(10,12), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,7)(6,8), (1,4)(2,6)(3,7)(5,8)\rangle$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $F_8$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $C_{28}$ (2)	$(C_2^4\times C_4)$ $\,\rtimes\,$ $C_7$	$(C_2^3\times C_4)$ $\,\rtimes\,$ $C_{14}$ (2)	$C_2^3$ $\,\rtimes\,$ $(C_2\times C_{28})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_{14}$	$(C_2^2\times F_8)$ . $C_2$	$(C_2\times F_8)$ . $C_2^2$	$C_2^2$ . $(C_2\times F_8)$	all 6

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

Homology

Abelianization:	$C_{2} \times C_{28} \simeq C_{2} \times C_{4} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 753 subgroups in 127 conjugacy classes, 24 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $F_8$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2\times C_{28}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times F_8$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_4$	$G/\operatorname{Fit} \simeq$ $C_7$
Radical:	$R \simeq$ $C_2^4:C_{28}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_{14}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4\times C_4$
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

Normal subgroups up to automorphism

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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 448: $C_2^4:C_{28}$

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

Order 112: $C_2\times F_8$ x 3

Order 64: $C_2^4\times C_4$

Order 56: $C_2\times C_{28}$, $F_8$

Order 32: $C_2^3\times C_4$ x 6, $C_2^5$

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

Order 16: $C_2^2\times C_4$ x 20, $C_2^4$ x 7

Order 14: $C_{14}$ x 3

Order 8: $C_2^3$ x 23, $C_2\times C_4$ x 18

Order 7: $C_7$

Order 4: $C_2^2$ x 23, $C_4$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 448: $C_2^4:C_{28}$

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

Order 112: $C_2\times F_8$ x 2

Order 64: $C_2^4\times C_4$

Order 56: $C_2\times C_{28}$, $F_8$

Order 32: $C_2^3\times C_4$ x 4, $C_2^5$

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

Order 16: $C_2^2\times C_4$ x 6, $C_2^4$ x 5

Order 14: $C_{14}$ x 2

Order 8: $C_2^3$ x 9, $C_2\times C_4$ x 6

Order 7: $C_7$

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 448: $C_2^4:C_{28}$ ($C_1$)

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

Order 112: $C_2\times F_8$ ($C_4$) x 2, $C_2\times F_8$ ($C_2^2$)

Order 64: $C_2^4\times C_4$ ($C_7$)

Order 56: $F_8$ ($C_2\times C_4$)

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

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

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

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

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

Order 1: $C_1$ ($C_2^4:C_{28}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 448: $C_2^4:C_{28}$ ($C_1$)

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

Order 112: $C_2\times F_8$ ($C_2^2$), $C_2\times F_8$ ($C_4$)

Order 64: $C_2^4\times C_4$ ($C_7$)

Order 56: $F_8$ ($C_2\times C_4$)

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

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

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

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

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

Order 1: $C_1$ ($C_2^4:C_{28}$)

Series

Derived series

$C_2^4:C_{28}$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^4:C_{28}$

$\rhd$

$C_2^2\times F_8$

$\rhd$

$C_2\times F_8$

$\rhd$

$F_8$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Lower central series

$C_2^4:C_{28}$

$\rhd$

$C_2^3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.