C₃×#72228227 #72228230

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₃×#72228227

Splittings

Fitting	C₂×C₃²×AGL(1,ℤ/16) ⋊ C₂
Sylow	C₃² ⋊ #7622637

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×D₃×#442
commutator	C₃×C₈	C₂⁴×C₃×C₄
Frattini	C₂×C₈	C₂⁴×C₃×D₃
Fitting	C₂×C₃²×AGL(1,ℤ/16)	C₂
socle	C₂×C₃²	C₂²×#442
prime radical	C₃×#11701722	C₂
cosocle	C₂×C₃×C₈	C₂⁵×C₃
hypercenter	C₂×C₃×AGL(1,ℤ/16)	D₃
hypocenter	C₃	C₃×#7622637
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97 ◁ C₂×C₃×AGL(1,ℤ/16)

8007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 38 points:

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

Conjugacy classes & character table

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

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

C3×#72228227 #72228230