C₂×#10710960 #72254292

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#10710960

Splittings

Sylow

C₃ ⋊ C₂×#12755

Subgroups

	subgroup	G/H
center	C₂²	#72142
commutator	C₂×C₃×D₄	C₂³×C₄
Frattini	C₂×#41	C₂³×D₃
Fitting, prime radical	C₂×C₃×#3537	C₂
socle	C₂²×C₃	#1790
cosocle	C₂×C₃×#41	C₂⁴
hypercenter	C₂×#3537	D₃
hypocenter	C₃	C₂×#12755
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3537

5598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(19,20,21);(5,12)(6,15)(9,14)(10,16)(11,17)(13,18);(3,5)(4,9,7,12)(6,13)(8,14)(10,11,16,18)(15,17)(20,21);(3,4)(5,11,14,18)(6,10)(7,8)(9,13,12,17)(15,16)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×#12755, #72254539, #72269887
maximal normal subgroup of	C₂×D₃×#12755, #72254539, #72269887

GAP

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

gap> G := Group((1,2)(19,20,21), (5,12)(6,15)(9,14)(10,16)(11,17)(13,18), (3,5)(4,9,7,12)(6,13)(8,14)(10,11,16,18)(15,17)(20,21), (3,4)(5,11,14,18)(6,10)(7,8)(9,13,12,17)(15,16));

C2×#10710960 #72254292