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C₂²×#672 #65385

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×#672

Splittings

C₅ ⋊ C₂²×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₅
commutator, hypocenter	C₅	C₂²×C₁₆
Frattini	C₈	C₂²×D₅
Fitting	C₂²×C₅×C₈	C₂
socle, prime radical	C₂³×C₅	C₈
cosocle	C₅×C₈	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₈

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2)(3,4)(6,7)(8,9)(10,11,12,15,13,16,18,21,14,17,19,22,20,23,24,25);(3,4)(5,6,8,9,7);(1,2)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598148, #10598162, #10598213, …
maximal normal subgroup of	#10598148, #10598162, #10598213, …
maximal quotient of	C₂×C₄×#672, C₂×C₄⋊#672, #10600537, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 723);