C₇×#5005 #13056041

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#5005

Splittings

Fitting	C₃×C₇×#117 ⋊ C₂
Hall	C₃×C₇ ⋊ #421
Sylow	C₃ ⋊ C₇×#421

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#117	C₂
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₇×#235	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#117	D₃
hypocenter	C₃	C₇×#421
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#117

308 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2,3,4,5,6,7)(8,9,10)(12,17,18,19)(13,20)(16,23,24,25)(21,26);(11,13)(12,16)(14,21)(15,20)(17,25)(18,24)(19,23)(22,26);(9,10)(11,12)(13,16)(14,17)(15,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5330);