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C₂×D₅₆ #6466

order	224
isoclinism	isoclinic class of #1132
automorphism	Aut(G) order 10752
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₅₆

Splittings

Fitting	C₂×C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂²	D₂₈
commutator, cosocle	C₄×C₇	C₂³
Frattini	C₄	C₂²×D₇
Fitting	C₂×C₇×C₈	C₂
socle	C₂²×C₇	D₄
hypercenter	C₂×C₈	D₇
hypocenter	C₇	C₂×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 17 points:

(3,4,6,8,9,7,5)(10,11,15,17,13,16,12,14);(1,2)(4,5)(6,7)(8,9)(11,14)(12,15)(16,17);(1,2)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#89286, #89290, #89304, …
maximal normal subgroup of	#89286, #89290, #89304, …
maximal quotient of	#89454, C₄×D₅₆, #89466, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 98);