C₃×#89562 #13055252

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

Aliases

direct product

C₃×#89562

Splittings

Hall	C₃×C₇ ⋊ C₄×Q₁₆
Sylow	C₇ ⋊ C₃×C₄×Q₁₆

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₇
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#103	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×#103	D₇
hypocenter	C₇	C₃×C₄×Q₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#103

220 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 1344

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

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4541);