C₂×#72296138 #72296491

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72296138

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁹	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

8168952 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)(19,20,22,21);(3,4,5,6)(7,8,9,10)(12,14)(15,16,17,18)(19,21,22,20);(11,12)(13,14);(15,16)(17,18);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72296156, C₂×#72296169, C₂×#72296173, …
maximal normal subgroup of	C₂×#72296156, C₂×#72296169, C₂×#72296173, …

GAP

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

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

C2×#72296138 #72296491