C₂×#2111324 #72325215

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

Aliases

direct product

C₂×#2111324

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

45755 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232172, C₂×#72232146, C₂×#72232163
maximal normal subgroup of	C₂×#72232172, C₂×#72232146, C₂×#72232163

GAP

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

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

C2×#2111324 #72325215