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C₄×D₅₆ #89464

order	448
isoclinism	isoclinic class of #1132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₄×D₅₆

Splittings

Fitting	C₄×C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ C₄×D₈

Subgroups

	subgroup	G/H
center	C₂×C₄	D₂₈
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×C₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₅₆	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×C₈	D₇
hypocenter	C₇	C₄×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2,4,6,7,5,3)(12,13,17,19,15,18,14,16);(2,3)(4,5)(6,7)(13,16)(14,17)(18,19);(8,9,10,11)

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725271, #11725277, #11725310, …
maximal normal subgroup of	#11725271, #11725277, #11725310, …
maximal quotient of	C₈×D₅₆, #11725806, #11725832, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 226);