C₂×D₃×#2263 #72255368

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×D₃×#2263

Splittings

Fitting	C₂×C₃×#2263 ⋊ C₂
Sylow	C₃ ⋊ C₂²×#2263

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₁₆
commutator, cosocle	C₃×C₁₆	C₂⁵
Frattini	C₁₆	C₂⁴×D₃
Fitting	C₂×C₃×#2263	C₂
socle	C₂²×C₃	C₂²×D₁₆
hypercenter	C₂×#2263	D₃
hypocenter	C₃	C₂²×#2263
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#2263

2948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 37 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×#13658
maximal normal subgroup of	C₂×D₃×#13658

GAP

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

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

C2×D3×#2263 #72255368