C₃×#89901 #13055561

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#89901

Splittings

Fitting	C₂×C₃×C₇×M₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #408
Sylow	C₇ ⋊ C₃×#408

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₃×#1157	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	D₇
hypocenter	C₇	C₃×#408
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

316 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 126 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)(12,14)(15,17,18,24,19,23,25,29)(16,21,22,28,20,26,27,30);(16,20)(17,23)(22,27)(24,29);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,21)(18,22)(19,20)(23,26)(24,28)(25,27)(29,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4850);