C₃×#89800 #13055461

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#89800

Splittings

Fitting	C₃×C₄×C₇×Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #464
Sylow	C₇ ⋊ C₃×#464

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₄×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₃×D₂₈	C₂²
hypercenter	C₃×C₄×Q₈	D₇
hypocenter	C₇	C₃×#464
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×Q₈

224 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 30
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

237 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4750);