C₇×#4912 #13055947

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

Aliases

direct product

C₇×#4912

Splittings

Fitting	C₃×C₇×D₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #508
Sylow	C₃ ⋊ C₇×#508

Subgroups

	subgroup	G/H
center	C₂×C₇	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³×C₇
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×D₁₆	C₂
socle	C₂×C₃×C₇	C₂×D₈
hypercenter	C₇×D₁₆	D₃
hypocenter	C₃	C₇×#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×D₁₆

168 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 42
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3,4,5,6,7)(8,9,10)(12,18)(13,22)(14,26)(15,27)(17,30)(19,36)(21,35)(23,24)(25,41)(28,42)(29,40)(31,38)(32,39)(33,37);(11,13)(12,17)(14,22)(15,29)(16,25)(18,21)(19,35)(20,38)(23,26)(24,27)(28,41)(30,33)(31,37)(32,34)(36,39)(40,42);(9,10)(11,12)(13,21)(14,19)(15,20)(16,18)(17,25)(22,30)(23,39)(24,38)(26,37)(27,34)(28,33)(29,32)(31,40)(35,41)(36,42)

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, 5236);