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C₄×#1055 #88042

order	416
isoclinism	isoclinic class of D₁₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

Splittings

C₁₃ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	D₁₃
commutator, hypocenter	C₁₃	C₄×C₈
Frattini	C₂×C₄	C₂×D₁₃
Fitting	C₄²×C₁₃	C₂
socle, prime radical	C₂²×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂²
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₄²

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#1055, #11718046, #11718047, …
maximal normal subgroup of	C₈×#1055, #11718046, #11718047, …
maximal quotient of	C₈×#1055, #11718047, C₄×#6090, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 9);