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C₄×C₃⋊D₂₁ #92723

order	504
isoclinism	isoclinic class of C₃⋊D₂₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₄×C₃⋊D₂₁

Splittings

Fitting	C₃²×C₄×C₇ ⋊ C₂
derived, Hall	C₃²×C₇ ⋊ C₂×C₄
Sylow	C₃² ⋊ C₄×D₇
Sylow	C₇ ⋊ C₄×C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₄	C₃⋊D₂₁
commutator, hypocenter	C₃²×C₇	C₂×C₄
Frattini	C₂	C₂×C₃⋊D₂₁
Fitting	C₃²×C₄×C₇	C₂
socle, cosocle	C₂×C₃²×C₇	C₂²
prime radical	C₂×C₃⋊D₂₁	C₂
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 252

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11759149, #11759152, #11759190, …
maximal normal subgroup of	#11759149, #11759152, #11759190, …
maximal quotient of	C₈×C₃⋊D₂₁, #11759304, C₄×C₃⋊Dic₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 149);