C₃×#89468 #13055129

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#89468

Splittings

Fitting	C₃×C₄×C₇×C₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #495
Sylow	C₇ ⋊ C₃×#495

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₄×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₃×D₅₆	C₂
hypercenter	C₃×C₄×C₈	D₇
hypocenter	C₇	C₃×#495
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈

260 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 94 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)(15,17,18,24,19,25,27,30)(16,21,22,23,20,28,26,29);(11,12,13,14)(16,20)(21,28)(22,26)(23,29);(5,6)(7,8)(9,10)(12,14)(15,16)(17,23)(18,26)(19,20)(21,24)(22,27)(25,29)(28,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4418);