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C₂³×#168 #66114

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×#168

Splittings

C₅ ⋊ C₂³×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₅
commutator, hypocenter	C₅	C₂³×C₈
Frattini	C₄	C₂³×D₅
Fitting	C₂³×C₄×C₅	C₂
socle, prime radical	C₂⁴×C₅	C₄
cosocle	C₄×C₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₄

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 320

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×#168, #10600240, #10600242, …
maximal normal subgroup of	C₂²⋊C₄×#168, #10600240, #10600242, …
maximal quotient of	C₂²×C₄×#168, C₂²×C₄⋊#168, C₂×#65214, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1452);