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C₂²×Dic₁₀ #4138

order	160
isoclinism	isoclinic class of Dic₁₀
automorphism	Aut(G) order 61440
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×Dic₁₀

Splittings

C₅ ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₅
commutator, cosocle	C₂×C₅	C₂⁴
Frattini	C₂	C₂³×D₅
Fitting	C₂²×C₄×C₅	C₂
socle, prime radical	C₂³×C₅	C₂²
hypercenter	C₂²×C₄	D₅
hypocenter	C₅	C₂²×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#64935, #64945, #64962, …
maximal normal subgroup of	#64935, #64945, #64962, …
maximal quotient of	C₂×C₄×Dic₁₀, C₂×#4015, C₂×#4016, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 213);