C₂²×#14190 #72337038

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#14190

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

79517 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#1258522, C₂²×#1418354, C₂²×#774408, …
maximal normal subgroup of	C₂²×#1258522, C₂²×#1418354, C₂²×#774408, …

GAP

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

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

C22×#14190 #72337038