D₃×#85453 #72269066

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

direct product

D₃×#85453

Splittings

Fitting

C₃×#568 ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂²×D₃×S₄
commutator, hypocenter	C₃×SL(2,𝔽₃)	C₂³×C₄
Fitting	C₃×#568	C₂×D₃
socle	C₂×C₃	C₂²×C₄×S₄
prime radical	D₃×#5923	C₂
cosocle	C₃×#41⋊C₃	C₂⁴
hypercenter	M₁₆	D₃×S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₄ ◁ M₁₆

2788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 51 points:

(1,2,3)(4,20,43,10,40,23)(5,32,42)(6,37,24,9,25,44)(7,26,51,21,8,39)(11,35,50)(12,34,45)(13,31,33)(14,47,15)(16,46,41)(17,18,36,19,28,48)(22,29,30)(27,49,38);(4,6,7,17,10,9,21,19)(5,11,13,29,12,27,15,16)(8,23,20,39,26,43,40,51)(14,33,32,45,31,47,34,42)(18,36,25,44,28,48,37,24)(22,30,38,49,41,46,50,35);(2,3);(4,5)(6,11)(7,13)(8,22)(9,27)(10,12)(14,24)(15,21)(16,19)(17,29)(18,33)(20,38)(23,30)(25,45)(26,41)(28,47)(31,44)(32,36)(34,48)(35,51)(37,42)(39,49)(40,50)(43,46)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11715001
maximal normal subgroup of	D₃×#11715001

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3)(4,20,43,10,40,23)(5,32,42)(6,37,24,9,25,44)(7,26,51,21,8,39)(11,35,50)(12,34,45)(13,31,33)(14,47,15)(16,46,41)(17,18,36,19,28,48)(22,29,30)(27,49,38), (4,6,7,17,10,9,21,19)(5,11,13,29,12,27,15,16)(8,23,20,39,26,43,40,51)(14,33,32,45,31,47,34,42)(18,36,25,44,28,48,37,24)(22,30,38,49,41,46,50,35), (2,3), (4,5)(6,11)(7,13)(8,22)(9,27)(10,12)(14,24)(15,21)(16,19)(17,29)(18,33)(20,38)(23,30)(25,45)(26,41)(28,47)(31,44)(32,36)(34,48)(35,51)(37,42)(39,49)(40,50)(43,46));

D3×#85453 #72269066