C₇×#4816 #13055848

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4816

Splittings

Fitting	C₃×C₇×#103 ⋊ C₂
Hall	C₃×C₇ ⋊ #470
Sylow	C₃ ⋊ C₇×#470

Subgroups

	subgroup	G/H
center	C₂²×C₇	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#103	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂²×D₃×C₇	D₄
hypercenter	C₇×#103	D₃
hypocenter	C₃	C₇×#470
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#103

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

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, 5137);