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C₂²×D₅₆ #90431

order	448
isoclinism	isoclinic class of #1132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×D₅₆

Splittings

Fitting	C₂²×C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ C₂²×D₈

Subgroups

	subgroup	G/H
center	C₂³	D₂₈
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₂²×C₇×C₈	C₂
socle	C₂³×C₇	D₄
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	C₂²×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726427, #11726428, #11726429, …
maximal normal subgroup of	#11726427, #11726428, #11726429, …
maximal quotient of	C₂×#89454, C₂×C₄×D₅₆, C₂×#89466, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1193);