C₂²×C₂²⋊C₄×#95 #72321980

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄³ · Schur multiplier (H₂) = C₂¹⁶×C₄²

Aliases

direct product

C₂²×C₂²⋊C₄×#95

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂²	C₂³×C₄³
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁸	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

820072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 1024

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#95×#120, C₂²×C₂²⋊C₄×#392, C₂²×#72278310
maximal normal subgroup of	C₂²×#95×#120, C₂²×C₂²⋊C₄×#392, C₂²×#72278310

GAP

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

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

C22×C22⋊C4×#95 #72321980