C₇×#5925 #13062101

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

solvable

Aliases

direct product

C₇×#5925

Splittings

Fitting

C₇×H(5,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂²×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂³×C₇
Frattini	C₂	C₂²×C₇×S₄
Fitting	C₇×H(5,𝔽₂)	D₃
hypercenter	C₇×Q₈	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₇×Q₈

292 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

161 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11390);