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C₃×Dic₂₆ #64611

order	312
isoclinism	isoclinic class of Dic₂₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

C₃×Dic₂₆

Splittings

Hall	C₃×C₁₃ ⋊ Q₈
Sylow	C₁₃ ⋊ C₃×Q₈

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×D₁₃
commutator, cosocle	C₂×C₁₃	C₂²×C₃
Frattini	C₂	C₂×C₃×D₁₃
Fitting	C₃×C₄×C₁₃	C₂
socle, prime radical	C₂×C₃×C₁₃	C₂²
hypercenter	C₃×C₄	D₁₃
hypocenter	C₁₃	C₃×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 312

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597599, #10597601, #10597604, …
maximal normal subgroup of	#10597599, #10597601, #10597604, …
maximal quotient of	C₃×#6101, C₃×C₄⋊Dic₁₃, C₂×C₃×Dic₂₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 27);