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C₂×#6104 #88185

order	416
isoclinism	isoclinic class of #6104
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₂×D₄×C₁₃ ⋊ C₂
Sylow	C₁₃ ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂²	C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₂×D₄×C₁₃	C₂
socle	C₂²×C₁₃	D₄
hypercenter	C₂×D₄	D₁₃
hypocenter	C₁₃	C₂×D₈
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718410, #11718417, #11718434, …
maximal normal subgroup of	#11718410, #11718417, #11718434, …
maximal quotient of	C₂×#88047, C₂×#88049, #11718628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 152);