C₂²×#14337 #72340082

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

direct product

C₂²×#14337

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

257317 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#517592, C₂×#72337387, C₂×#72337388
maximal normal subgroup of	C₂²×#517592, C₂×#72337387, C₂×#72337388
maximal quotient of	C₂×#72337388, C₂×#72337370

GAP

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

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

C22×#14337 #72340082