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C₂²×#1055 #88174

order	416
isoclinism	isoclinic class of D₁₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₁₃ ⋊ C₂²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₁₃
commutator, hypocenter	C₁₃	C₂²×C₈
Frattini	C₄	C₂²×D₁₃
Fitting	C₂²×C₄×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₄
cosocle	C₄×C₁₃	C₂³
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 416

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718116, #11718121, #11718142, …
maximal normal subgroup of	#11718116, #11718121, #11718142, …
maximal quotient of	C₂×C₄×#1055, C₂×C₄⋊#1055, #11718586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 141);