C₂²×#14681 #72290832

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂²×#14681

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

430133 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#6342374, C₂²×#4307937, C₂²×#517411, …
maximal normal subgroup of	C₂²×#6342374, C₂²×#4307937, C₂²×#517411, …
maximal quotient of	C₂×#72287888, C₂²×#154077, C₂×#72339322, …

GAP

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

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

C22×#14681 #72290832