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C₂²⋊#1057 #88120

order	416
isoclinism	isoclinic class of #6118
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

Splittings

C₁₃ ⋊ C₂²⋊C₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×#261
commutator	C₂×C₁₃	C₂×C₈
Fitting, prime radical	C₂³×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄
cosocle	C₂×Dic₁₃	C₂²
hypercenter	C₂³	#261
hypocenter	C₁₃	C₂²⋊C₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 208

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

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

Polycyclic presentation:

2^5,13|a^2=c,c^2=e,b^a=bd,f^a=f^8,f^c=f^12

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718285, #11718286, #11718287, …
maximal normal subgroup of	#11718285, #11718286, #11718287, …
maximal quotient of	C₂²⋊#6092, #11718284, #11718285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 87);