C₂×#483326 #72231981

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#483326

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#106
commutator	C₈	C₂⁴×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂²×#41	C₄⋊C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ C₂×C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 42 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224974, C₂×#72216441, C₂×#72216726, …
maximal normal subgroup of	C₂×#72224974, C₂×#72216441, C₂×#72216726, …
maximal quotient of	#72231984, #72234315, C₂×D₄×#1368

GAP

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

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

C2×#483326 #72231981