C₇×#568⋊C₃ #13060761

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₇ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product	C₇×#568⋊C₃
unambiguous semidirect product	#568⋊(C₃×C₇)

Splittings

Fitting, Hall	C₇×#568 ⋊ C₃
Sylow	#568 ⋊ C₃×C₇

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂²×A₄
commutator, hypocenter	Q₈	C₂×C₃×C₄×C₇
Frattini	C₄	C₂²×C₇×A₄
Fitting	C₇×#568	C₃
socle	C₂×C₇	C₂×C₄×A₄
prime radical	C₂×C₇×#41⋊C₃	C₂
cosocle	#41	C₂²×C₃×C₇
hypercenter	C₇×M₁₆	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₄×C₇ ◁ C₇×M₁₆

146 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,10,15,13,14,20,25)(11,17,32,28,24,31,33,37,21,16,19,38,22,27,34,36,12,30,18,29,23,26,35,39);(8,21,13,23)(9,31,14,30)(10,32,20,34)(11,24,22,12)(15,39,25,38)(16,17,26,27)(18,35,33,19)(28,29,36,37)

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10050);