C₂×#11797877 #72269713

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product

C₂×#11797877

Splittings

Fitting	C₂×C₃²×#346 ⋊ C₂
Sylow	C₃² ⋊ C₂×#2179

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#3745
commutator	C₃²×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×#346	C₂
socle	C₂²×C₃²	C₄×D₈
prime radical	C₂×#10592603	C₂
cosocle	C₂×C₃²×C₈	C₂⁴
hypercenter	C₂×#346	C₃⋊D₃
hypocenter	C₃²	C₂×#2179
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

1962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 4 generators on 40 points:

(6,7,8)(9,12,14,13,15,22,30,31)(10,17,18,37,20,27,19,29)(11,23,25,32,26,28,40,24)(16,33,34,39,36,21,35,38);(3,4,5)(9,11,22,23,14,25,31,32,15,26,12,28,30,40,13,24)(10,16,29,39,19,35,27,33,20,36,37,38,18,34,17,21);(1,2);(4,5)(7,8)(9,10,15,20)(11,21,26,33)(12,27,22,17)(13,29,31,37)(14,18,30,19)(16,32,36,24)(23,34,28,35)(25,38,40,39)

Conjugacy classes & character table

384 conjugacy classes. The character table is too large to display.

GAP

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

gap> G := Group((6,7,8)(9,12,14,13,15,22,30,31)(10,17,18,37,20,27,19,29)(11,23,25,32,26,28,40,24)(16,33,34,39,36,21,35,38), (3,4,5)(9,11,22,23,14,25,31,32,15,26,12,28,30,40,13,24)(10,16,29,39,19,35,27,33,20,36,37,38,18,34,17,21), (1,2), (4,5)(7,8)(9,10,15,20)(11,21,26,33)(12,27,22,17)(13,29,31,37)(14,18,30,19)(16,32,36,24)(23,34,28,35)(25,38,40,39));

C2×#11797877 #72269713