C₂²×#34190 #72325825

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#34190

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

111277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7631563, C₂²×#7653121, C₂²×#7626568, …
maximal normal subgroup of	C₂²×#7631563, C₂²×#7653121, C₂²×#7626568, …
maximal quotient of	C₂²×#520468, C₂²×#513714

GAP

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

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

C22×#34190 #72325825