C₂×#11797972 #72227377

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#11797972

Splittings

Sylow

C₃² ⋊ C₂×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center	C₂²	#10590346
commutator	C₃²×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂²×D₃²
Fitting	C₂×C₃²×#97	C₂²
socle	C₂²×C₃²	#442
prime radical	C₂×#10590634	C₂
cosocle	C₂×C₃²×C₈	C₂⁴
hypercenter	C₂×#97	D₃²
hypocenter	C₃²	C₂×AGL(1,ℤ/16)
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97

1724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

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

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

C2×#11797972 #72227377