C₉×#19745 #72244452

order	2304
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₉×#19745

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×#127
commutator	C₄²	C₂⁴×C₉
Frattini, cosocle	C₃×C₄²	C₂⁴×C₃
socle	C₂²×C₃	C₂×C₃×#127
prime radical	C₃×#19745	C₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

1491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 4 generators on 25 points:

(1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(14,16)(17,18,19,20,21,22,23,24,25);(2,6)(3,4)(7,8)(10,14)(11,15)(13,16);(1,3)(4,5)(6,7)(10,13)(11,15)(14,16);(9,10,15,14,12,16,11,13)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	D₉×#19745
maximal subgroup of	D₉×#19745, 3₋¹⁺²×#19745
maximal normal subgroup of	D₉×#19745, 3₋¹⁺²×#19745

GAP

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

gap> G := Group((1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(14,16)(17,18,19,20,21,22,23,24,25), (2,6)(3,4)(7,8)(10,14)(11,15)(13,16), (1,3)(4,5)(6,7)(10,13)(11,15)(14,16), (9,10,15,14,12,16,11,13));

C9×#19745 #72244452