C₂×#72240113 #72289562

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂×#72240113

Subgroups

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

8281884 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72286701, C₂×#72293994, C₂×#72296156, …
maximal normal subgroup of	C₂×#72286701, C₂×#72293994, C₂×#72296156, …
maximal quotient of	C₂²×#72240113

GAP

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

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

C2×#72240113 #72289562