C₃×#11925650 #72229709

order	3600
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#11925650

Splittings

Hall	C₃²×C₅² ⋊ C₂×Q₈
Sylow	C₅² ⋊ C₂×C₃×Dic₆
Sylow	C₃² ⋊ #87872

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×D₁₅⋊D₅
commutator, cosocle	C₂×C₃×C₅²	C₂³×C₃
Frattini	C₂	C₂×C₃×D₁₅⋊D₅
Fitting	C₃²×C₄×C₅²	C₂²
socle	C₂×C₃²×C₅²	C₂³
prime radical	C₂×C₃²×C₅⋊D₅	C₂²
hypercenter	C₃×C₄	D₁₅⋊D₅
hypocenter	C₃×C₅²	C₂×C₃×Q₈
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

432 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

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

GAP

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

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