C₂²×#34494 #72299872

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#34494

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#550	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

95057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

(5,6,7,8)(10,13)(14,16)(17,18)(19,22)(20,24)(21,23);(9,10,12,13)(11,14,15,16)(17,19,21,20)(18,22,23,24);(11,15)(14,16)(17,19,21,20)(18,22,23,24);(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,22,23,24);(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23);(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×#7878926, C₂²×C₄×#3543
maximal normal subgroup of	C₂²×#7878926, C₂²×C₄×#3543

GAP

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

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

C22×#34494 #72299872