C₃×#10948637 #72227362

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#10948637

Splittings

Fitting	C₂×C₃²×#444 ⋊ C₂
Sylow	C₃² ⋊ #33468

Subgroups

	subgroup	G/H
center	C₃×C₄	#5014
commutator	C₃×C₈	C₂³×C₃×C₄
Frattini	C₂×C₈	C₂³×C₃×D₃
Fitting	C₂×C₃²×#444	C₂
socle	C₂×C₃²	C₂×#442
prime radical	C₃×#83894	C₂
cosocle	C₂×C₃×C₈	C₂⁴×C₃
hypercenter	C₂×C₃×#444	D₃
hypocenter	C₃	C₃×#33468
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97 ◁ C₂×C₃×#444

1536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2,3)(4,5,6)(8,16)(9,20)(15,17)(18,32)(21,23)(22,37)(25,31)(26,38);(8,15,16,17)(11,27)(14,29,30,33)(18,25,32,31)(19,35,36,34)(22,37)(24,28)(26,38);(5,6)(7,9)(8,14,16,30)(10,23)(11,26)(12,21)(13,20)(15,29,17,33)(18,34,32,35)(19,31,36,25)(22,28)(24,37)(27,38);(7,8,10,17,13,16,12,15)(9,19,21,34,20,36,23,35)(11,25,24,32,27,31,28,18)(14,26,29,22,30,38,33,37)

Conjugacy classes & character table

270 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×#72228227
maximal normal subgroup of	C₃×#72228227

GAP

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

gap> G := Group((1,2,3)(4,5,6)(8,16)(9,20)(15,17)(18,32)(21,23)(22,37)(25,31)(26,38), (8,15,16,17)(11,27)(14,29,30,33)(18,25,32,31)(19,35,36,34)(22,37)(24,28)(26,38), (5,6)(7,9)(8,14,16,30)(10,23)(11,26)(12,21)(13,20)(15,29,17,33)(18,34,32,35)(19,31,36,25)(22,28)(24,37)(27,38), (7,8,10,17,13,16,12,15)(9,19,21,34,20,36,23,35)(11,25,24,32,27,31,28,18)(14,26,29,22,30,38,33,37));

C3×#10948637 #72227362