C₂²×#13791 #72339472

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#13791

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁴×C₄²
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

69901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#540671, C₂²×#1192577, C₂²×#525856
maximal normal subgroup of	C₂²×#540671, C₂²×#1192577, C₂²×#525856

GAP

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

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

C22×#13791 #72339472