C₂×#150142 #72322644

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#150142

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#357
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂³×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 42 points:

(12,18)(13,17)(14,24)(15,26)(19,36)(20,37)(21,35)(22,32)(23,30)(25,42)(27,41)(28,40)(29,34)(31,33)(38,39);(4,8)(9,10);(3,4,6,10,7,8,5,9)(11,12,13,19,16,21,17,31)(14,25,22,39,27,34,28,37)(15,29,23,38,30,42,26,20)(18,32,33,24,35,40,36,41);(1,2)(3,5,7,6)(4,9,8,10)(11,13,16,17)(12,19,21,31)(14,22,27,28)(15,23,30,26)(18,33,35,36)(20,29,38,42)(24,40,41,32)(25,39,34,37)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322687, #72323020, #72323054
maximal normal subgroup of	#72322687, #72323020, #72323054
maximal quotient of	#72322654

GAP

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

gap> G := Group((12,18)(13,17)(14,24)(15,26)(19,36)(20,37)(21,35)(22,32)(23,30)(25,42)(27,41)(28,40)(29,34)(31,33)(38,39), (4,8)(9,10), (3,4,6,10,7,8,5,9)(11,12,13,19,16,21,17,31)(14,25,22,39,27,34,28,37)(15,29,23,38,30,42,26,20)(18,32,33,24,35,40,36,41), (1,2)(3,5,7,6)(4,9,8,10)(11,13,16,17)(12,19,21,31)(14,22,27,28)(15,23,30,26)(18,33,35,36)(20,29,38,42)(24,40,41,32)(25,39,34,37));

C2×#150142 #72322644