C₂×#7683363 #72298706

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7683363

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

124061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 18 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233771
maximal normal subgroup of	C₂×#72233771
maximal quotient of	C₂×#72298085, C₂×#72298100, C₂×#72298082

GAP

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

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

C2×#7683363 #72298706