C₂×#2110323 #72294675

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#2110323

Subgroups

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

61085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293942, C₂×#72230859, C₂×#72277929, …
maximal normal subgroup of	C₂×#72293942, C₂×#72230859, C₂×#72277929, …
maximal quotient of	C₂×#72293968, C₂×#72293973, C₂×#72278047

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

C2×#2110323 #72294675