C₃×#88132 #11928768

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#88132

Splittings

Hall	C₃×C₁₃ ⋊ C₂²⋊Q₈
Sylow	C₁₃ ⋊ C₃×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₁₃
Fitting	C₃×C₁₃×C₂²⋊C₄	C₂
socle	C₂²×C₃×C₁₃	C₂³
prime radical	C₂³×C₃×C₁₃	C₂²
hypercenter	C₃×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	C₃×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊C₄

148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 28
transitive degree	μ_t(G) = 624

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3)(4,5,7,9,11,13,15,16,14,12,10,8,6)(18,20)(21,23,24,28)(22,25,26,27);(17,18)(19,20);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(21,22,24,26)(23,27,28,25)

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 880);