C₂×#72335134 #72335220

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂×#72335134

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

965480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

(3,4)(5,6)(7,8,9,10)(12,14)(17,21)(20,22);(8,10)(11,12)(13,14)(16,19)(20,22);(3,4,5,6)(7,8)(9,10)(12,14)(15,16)(17,22)(18,19)(20,21);(4,6)(11,12)(13,14);(1,2);(15,17)(16,20)(18,21)(19,22);(3,4)(5,6)(7,8)(9,10)(15,16)(17,22)(18,19)(20,21)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72335137
maximal normal subgroup of	C₂×#72335137

GAP

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

gap> G := Group((3,4)(5,6)(7,8,9,10)(12,14)(17,21)(20,22), (8,10)(11,12)(13,14)(16,19)(20,22), (3,4,5,6)(7,8)(9,10)(12,14)(15,16)(17,22)(18,19)(20,21), (4,6)(11,12)(13,14), (1,2), (15,17)(16,20)(18,21)(19,22), (3,4)(5,6)(7,8)(9,10)(15,16)(17,22)(18,19)(20,21));

C2×#72335134 #72335220