C₇×#4720 #13055756

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#4720

Splittings

Hall	C₃×C₇ ⋊ #433
Sylow	C₃ ⋊ C₇×#433

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×C₂²⋊C₈	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×C₂²⋊C₈	D₃
hypocenter	C₃	C₇×#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×C₂²⋊C₈

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

normal subgroups only

This group has 192 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) = 34
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,4,5,6,7)(8,9,10)(13,17)(15,18)(19,21,22,27,23,28,29,33)(20,25,26,32,24,30,31,34);(11,13)(12,15)(14,17)(16,18)(20,24)(25,30)(26,31)(32,34);(9,10)(11,12,14,16)(13,15,17,18)(19,20)(21,25)(22,26)(23,24)(27,32)(28,30)(29,31)(33,34)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5045);