Finite Group Databasebeta

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C₄×Dic₂₁ #66628

order	336
isoclinism	isoclinic class of D₂₁
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

C₄×Dic₂₁

Splittings

derived, Hall	C₃×C₇ ⋊ C₄²
Sylow	C₇ ⋊ C₄×Dic₃
Sylow	C₃ ⋊ C₄×Dic₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₂₁
commutator, hypocenter	C₃×C₇	C₄²
Frattini	C₂²	C₂×D₂₁
Fitting	C₂×C₃×C₄×C₇	C₂
socle, prime radical, cosocle	C₂²×C₃×C₇	C₂²
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620486, #10620489, #10620511, …
maximal normal subgroup of	#10620486, #10620489, #10620511, …
maximal quotient of	C₄×C₃⋊(C₇⋊C₈), #10620642, C₈×Dic₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 97);