C₁₃×#942 #11928920

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#942

Splittings

Fitting	C₂×C₃×Q₈×C₁₃ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #137
Sylow	C₃ ⋊ C₁₃×#137

Subgroups

	subgroup	G/H
center	C₂×C₁₃	C₂×#67
commutator, cosocle	C₃×C₄	C₂³×C₁₃
Frattini	C₄	C₂²×D₃×C₁₃
Fitting	C₂×C₃×Q₈×C₁₃	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
prime radical	C₁₃×#235	C₂
hypercenter	C₂×Q₈×C₁₃	D₃
hypocenter	C₃	C₁₃×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×Q₈×C₁₃

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 32
transitive degree	μ_t(G) = 624

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16)(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(15,16)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 1032);