C₂²×#13868 #72296089

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

Aliases

direct product

C₂²×#13868

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

186149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#520493, C₂²×#612434, C₂²×#521312, …
maximal normal subgroup of	C₂²×#520493, C₂²×#612434, C₂²×#521312, …
maximal quotient of	C₂²×#154095, C₂×#72337367

GAP

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

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

C22×#13868 #72296089