C₇×#4851 #13055867

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#4851

Splittings

Hall	C₃×C₇ ⋊ #475
Sylow	C₃ ⋊ C₇×#475

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₇×C₄⋊C₈	D₃
hypocenter	C₃	C₇×#475
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₄⋊C₈

192 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 1344

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5156);