C₂²×#513587 #72320953

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄² · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂²×#513587

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂²	C₂⁵×C₄²
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×#120	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

819848 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72320249
maximal normal subgroup of	C₂²×#72320249
maximal quotient of	C₂²×#95×#120

GAP

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

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

C22×#513587 #72320953