C₇×#4918 #13055953

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

Aliases

direct product

C₇×#4918

Splittings

Fitting	C₃×C₇×Q₃₂ ⋊ C₂
Hall	C₃×C₇ ⋊ #510
Sylow	C₃ ⋊ C₇×#510

Subgroups

	subgroup	G/H
center	C₂×C₇	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³×C₇
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×Q₃₂	C₂
socle	C₂×C₃×C₇	C₂×D₈
prime radical	C₇×#919	C₂
hypercenter	C₇×Q₃₂	D₃
hypocenter	C₃	C₇×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×Q₃₂

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

normal subgroups only

This group has 164 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)(11,13,16,21)(12,17,20,32)(14,26,29,42)(15,31,27,24)(18,36,37,41)(19,38,30,34)(22,40,39,25)(23,35,33,28);(11,12,16,20)(13,22,21,39)(14,25,29,40)(15,30,27,19)(17,33,32,23)(18,35,37,28)(24,41,31,36)(26,38,42,34);(9,10)(13,21)(17,32)(22,39)(23,33)(24,31)(26,42)(34,38)(36,41)

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5242);