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C₄⋊#1055 #88044

order	416
isoclinism	isoclinic class of Dic₂₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

Splittings

C₁₃ ⋊ C₄⋊C₈

Subgroups

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

38 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:

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

Polycyclic presentation:

2^5,13|a^2=c,b^2=d,c^2=e,b^a=bd,f^a=f^12

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718048, #11718049, #11718050, …
maximal normal subgroup of	#11718048, #11718049, #11718050, …
maximal quotient of	#11718049, #11718050, C₄⋊#6090, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 11);