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C₂³×Dic₁₃ #88258

order	416
isoclinism	isoclinic class of D₁₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×Dic₁₃

Splittings

C₁₃ ⋊ C₂³×C₄

Subgroups

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

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×Dic₁₃, #11718603, #11718604, …
maximal normal subgroup of	C₂²⋊C₄×Dic₁₃, #11718603, #11718604, …
maximal quotient of	C₂³×#1055, C₂²×#6099, C₂²×C₄×Dic₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 225);