C₂²×#14152 #72296045

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#14152

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

101621 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)(9,10)(11,12)(13,14,16,15)(17,18,21,23)(19,20,22,24);(6,8)(10,11)(13,14,16,15);(17,19)(18,20)(21,22)(23,24);(5,6,7,8)(9,11,12,10)(13,15,16,14)(17,20)(18,22)(19,23)(21,24);(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×#763814, C₂²×#4122655, C₂²×#612810, …
maximal normal subgroup of	C₂²×#763814, C₂²×#4122655, C₂²×#612810, …
maximal quotient of	C₂²×#154255

GAP

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

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

C22×#14152 #72296045