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C₂²×Dic₂₆ #88245

order	416
isoclinism	isoclinic class of Dic₂₆
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×Dic₂₆

Splittings

C₁₃ ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁴
Frattini	C₂	C₂³×D₁₃
Fitting	C₂²×C₄×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²
hypercenter	C₂²×C₄	D₁₃
hypocenter	C₁₃	C₂²×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718307, #11718317, #11718334, …
maximal normal subgroup of	#11718307, #11718317, #11718334, …
maximal quotient of	C₂×C₄×Dic₂₆, C₂×#88123, C₂×#88124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 212);