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C₂×#6106 #88195

order	416
isoclinism	isoclinic class of #6104
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₂×Q₈×C₁₃ ⋊ C₂
Sylow	C₁₃ ⋊ C₂×SD₁₆

Subgroups

	subgroup	G/H
center	C₂²	C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₂×Q₈×C₁₃	C₂
socle	C₂²×C₁₃	D₄
prime radical	C₂×D₅₂	C₂
hypercenter	C₂×Q₈	D₁₃
hypocenter	C₁₃	C₂×SD₁₆
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

68 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:

(3,4,6,8)(5,10,9,7)(11,12,14,16,18,20,22,23,21,19,17,15,13);(4,7)(5,9)(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	#11718415, #11718449, #11718467, …
maximal normal subgroup of	#11718415, #11718449, #11718467, …
maximal quotient of	C₂×#88048, C₂×#88049, #11718629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 162);