C₇×#4717 #13055752

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

Aliases

direct product

C₇×#4717

Splittings

Fitting	C₃×C₇×#97 ⋊ C₂
Hall	C₃×C₇ ⋊ #444
Sylow	C₃ ⋊ C₇×#444

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#97	C₂
socle	C₂×C₃×C₇	C₄×D₄
prime radical	C₇×#904	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#97	D₃
hypocenter	C₃	C₇×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₇×#97

208 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 94 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)(11,14,15,22,16,23,24,13)(12,17,18,25,20,26,19,21);(11,13,15,14,16,22,24,23)(12,17,19,21,20,26,18,25);(9,10)(11,12)(13,21)(14,17)(15,18)(16,20)(19,24)(22,25)(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, 5041);