C₇×#5045 #13056080

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#5045

Splittings

Hall	C₃×C₇ ⋊ #485
Sylow	C₃ ⋊ C₇×#485

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
hypercenter	C₇×C₂²⋊Q₈	D₃
hypocenter	C₃	C₇×#485
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×C₂²⋊Q₈

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

normal subgroups only

This group has 192 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) = 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)(19,22,24,30)(20,26,27,34)(21,28,23,32)(25,33,31,29);(11,13)(12,15)(14,17)(16,18)(19,21,24,23)(20,25,27,31)(22,28,30,32)(26,33,34,29);(9,10)(11,12,14,16)(13,15,17,18)(19,20)(21,25)(22,29)(23,31)(24,27)(26,28)(30,33)(32,34)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5369);