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C₂²×Q₈×D₅ #66277

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×Q₈×D₅

Splittings

Fitting	C₂²×C₅×Q₈ ⋊ C₂
Sylow	C₅ ⋊ C₂³×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂⁵
Frattini	C₂	C₂⁴×D₅
Fitting	C₂²×C₅×Q₈	C₂
socle	C₂³×C₅	C₂³
prime radical	C₂³×D₅	C₂²
hypercenter	C₂²×Q₈	D₅
hypocenter	C₅	C₂³×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600906, #10601059, #10604973, …
maximal normal subgroup of	#10600906, #10601059, #10604973, …
maximal quotient of	C₂²×#4033, C₂²×#4035, C₂²×#4034, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1615);