C₇×#5119 #13056156

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

Aliases

direct product

C₇×#5119

Splittings

Hall	C₃×C₇ ⋊ #424
Sylow	C₃ ⋊ C₇×#424

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂×C₇×M₁₆	D₃
hypocenter	C₃	C₇×#424
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆

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

normal subgroups only

This group has 228 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:

(1,2,3,4,5,6,7)(8,9,10)(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);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5445);