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C₄×#1057 #88108

order	416
isoclinism	isoclinic class of #261
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

Splittings

C₁₃ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#261
commutator, hypocenter	C₁₃	C₄×C₈
Frattini	C₂²	C₂×#261
Fitting	C₂×C₄×C₁₃	C₄
socle, prime radical	C₂²×C₁₃	C₂×C₄
cosocle	C₂×Dic₁₃	C₂²
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂×C₄

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 25 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718243, #11718244, C₈×#1057, …
maximal normal subgroup of	#11718243, #11718244, C₈×#1057, …
maximal quotient of	C₄×#6092, #11718231, C₈×#1057, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 75);