C₂²×#7641850 #72337196

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵×C₄

Aliases

direct product

C₂²×#7641850

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1619640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72235095
maximal normal subgroup of	C₂²×#72235095

GAP

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

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

C22×#7641850 #72337196