C₂²×#15404 #72339451

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#15404

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

(6,8)(13,14,15,16)(17,18,21,24)(19,20,23,22);(3,4);(10,12)(13,14)(15,16)(17,19,21,23)(18,22,24,20);(1,2)(3,4);(9,10)(11,12)(17,19)(18,22)(20,24)(21,23);(5,6,7,8)(17,20,21,22)(18,23,24,19)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#2814158, C₂²×#1420611, C₂²×#647923
maximal normal subgroup of	C₂²×#2814158, C₂²×#1420611, C₂²×#647923

GAP

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

gap> G := Group((6,8)(13,14,15,16)(17,18,21,24)(19,20,23,22), (3,4), (10,12)(13,14)(15,16)(17,19,21,23)(18,22,24,20), (1,2)(3,4), (9,10)(11,12)(17,19)(18,22)(20,24)(21,23), (5,6,7,8)(17,20,21,22)(18,23,24,19));

C22×#15404 #72339451