C₂×#10715811 #72255373

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

Aliases

direct product

C₂×#10715811

Splittings

Fitting	C₂×C₃×#1426 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13658

Subgroups

	subgroup	G/H
center	C₂²	#69339
commutator	C₃×C₁₆	C₂³×C₄
Frattini	C₂×C₁₆	C₂³×D₃
Fitting	C₂×C₃×#1426	C₂
socle	C₂²×C₃	#2177
prime radical	C₂×#69373	C₂
cosocle	C₂×C₃×C₁₆	C₂⁴
hypercenter	C₂×#1426	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×#1426

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

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

C2×#10715811 #72255373