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C₄×D₃×C₁₇ #87984

order	408
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×D₃×C₁₆ (order 384) · Inn(G) = D₃ · Out(G) = C₂²×C₁₆
homology	Abelianization (H₁) = C₂×C₄×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₃×C₁₇

Splittings

Fitting	C₃×C₄×C₁₇ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₄×C₁₇
Hall	C₃×C₁₇ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₁₇	D₃
commutator, hypocenter	C₃	C₂×C₄×C₁₇
Frattini	C₂	C₂×D₃×C₁₇
Fitting	C₃×C₄×C₁₇	C₂
socle	C₂×C₃×C₁₇	C₂²
prime radical	C₂×D₃×C₁₇	C₂
cosocle	C₂×C₃	C₂²×C₁₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₁₇

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 204

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

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3622, #11717741, D₃×C₈×C₁₇, …
maximal normal subgroup of	D₃×#3622, #11717741, D₃×C₈×C₁₇, …
maximal quotient of	D₃×C₈×C₁₇, C₁₇×ΓL(1,𝔽₂₅), C₄×Dic₃×C₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(408, 21);