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C₃×C₂²⋊#595 #11717843

order	816
isoclinism	isoclinic class of #63243
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃×C₂²⋊#595
unambiguous semidirect product	(C₂×#595)⋊(C₂×C₃)

Splittings

Fitting	C₂²×C₃×C₁₇ ⋊ C₄
Hall	C₃×C₁₇ ⋊ C₂²⋊C₄
Sylow	C₁₇ ⋊ C₃×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×#595
commutator	C₂×C₁₇	C₂×C₃×C₄
Frattini	C₂	C₂×C₃×#595
Fitting	C₂²×C₃×C₁₇	C₄
socle	C₂×C₃×C₁₇	C₂×C₄
prime radical	C₂²×C₃×D₁₇	C₂
cosocle	C₂×D₁₇	C₂²×C₃
hypercenter	C₂²×C₃	#595
hypocenter	C₁₇	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

68 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:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13108909, #13108910, C₃×#10587539, …
maximal normal subgroup of	#13108909, #13108910, C₃×#10587539, …
maximal quotient of	C₃×#10587539, C₃×#10587543, C₃×#10587545, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(816, 113);