C₇×#4755 #13055789

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

Aliases

direct product

C₇×#4755

Splittings

Fitting	C₃×C₇×#101 ⋊ C₂
Hall	C₃×C₇ ⋊ #473
Sylow	C₃ ⋊ C₇×#473

Subgroups

	subgroup	G/H
center	C₂×C₇	#884
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#101	C₂
socle	C₂×C₃×C₇	#121
prime radical	C₇×#235	C₂²
hypercenter	C₇×#101	D₃
hypocenter	C₃	C₇×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#101

200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 70 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)(11,14,16,27)(12,18,20,33)(13,21,23,25)(15,28,29,26)(17,30,32,41)(19,34,35,24)(22,38,39,37)(31,36,42,40);(11,13,15,22,16,23,29,39)(12,17,19,31,20,32,35,42)(14,25,26,38,27,21,28,37)(18,30,24,40,33,41,34,36);(9,10)(11,12)(13,17)(14,24)(15,19)(16,20)(18,28)(21,36)(22,31)(23,32)(25,40)(26,33)(27,34)(29,35)(30,37)(38,41)(39,42)

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, 5078);