C₂×D₃×#1419 #72255374

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×D₃×#1419

Splittings

Fitting	C₂×C₃×#1419 ⋊ C₂
Sylow	C₃ ⋊ C₂²×#1419

Subgroups

	subgroup	G/H
center	C₂²	D₃×#357
commutator	C₃×C₁₆	C₂³×C₄
Frattini	C₂×C₁₆	C₂³×D₃
Fitting	C₂×C₃×#1419	C₂
socle	C₂²×C₃	C₂×#357
prime radical	C₂²×D₃×D₁₆	C₂
cosocle	C₂×C₃×C₁₆	C₂⁴
hypercenter	C₂×#1419	D₃
hypocenter	C₃	C₂²×#1419
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#1419

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 37 points:

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

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

C2×D3×#1419 #72255374