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D₅×M₃₂ #65195

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

D₅×M₃₂

Splittings

Fitting	C₅×M₃₂ ⋊ C₂
Sylow	C₅ ⋊ C₂×M₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂²×D₅
commutator, socle	C₂×C₅	C₂²×C₈
Fitting	C₅×M₃₂	C₂
prime radical	C₂²×D₅	C₈
cosocle	C₅×C₈	C₂³
hypercenter	M₃₂	D₅
hypocenter	C₅	C₂×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ M₃₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598670, #10598671, #10598672, …
maximal normal subgroup of	#10598670, #10598671, #10598672, …
maximal quotient of	#10599861, D₅×#346, #10599864, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 533);