C₂×#10715809 #72255379

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10715809

Splittings

Fitting	C₂×C₃×#1418 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13658

Subgroups

	subgroup	G/H
center	C₂²	#69177
commutator	C₃×C₁₆	C₂³×C₄
Frattini	C₂×C₁₆	C₂³×D₃
Fitting	C₂×C₃×#1418	C₂
socle	C₂²×C₃	#2177
prime radical	C₂×#71509	C₂
cosocle	C₂×C₃×C₁₆	C₂⁴
hypercenter	C₂×#1418	D₃
hypocenter	C₃	C₂×#13658
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#1418

900 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)(7,13,16,29)(8,18)(12,24,27,26)(14,30,32,25)(15,28,33,31)(17,34)(20,35)(36,37);(1,2,3)(7,12,16,27)(8,17)(10,23)(11,22)(13,24,29,26)(14,15,32,33)(18,34)(19,21)(20,37)(25,31,30,28)(35,36);(4,5);(6,7,8,14,19,31,37,26,11,13,20,25,10,15,17,27,9,16,18,32,23,28,36,24,22,29,35,30,21,33,34,12)

Conjugacy classes & character table

120 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)(7,13,16,29)(8,18)(12,24,27,26)(14,30,32,25)(15,28,33,31)(17,34)(20,35)(36,37), (1,2,3)(7,12,16,27)(8,17)(10,23)(11,22)(13,24,29,26)(14,15,32,33)(18,34)(19,21)(20,37)(25,31,30,28)(35,36), (4,5), (6,7,8,14,19,31,37,26,11,13,20,25,10,15,17,27,9,16,18,32,23,28,36,24,22,29,35,30,21,33,34,12));

C2×#10715809 #72255379