C₇×#5431 #13057151

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₇×#5431

Splittings

Fitting

C₇×2₋¹⁺⁴ ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	GL(2,ℤ/4)
commutator, cosocle	#41⋊C₃	C₂²×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₇×2₋¹⁺⁴	D₃
hypercenter	C₇×D₄	S₄
hypocenter	SL(2,𝔽₃)	C₇×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×D₄

144 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 26 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)(8,11,23)(9,26,31,14,16,38)(10,34,36,19,20,37)(13,24,25)(15,32,39,18,28,33)(17,27)(21,35)(29,30);(8,9)(10,14)(11,17)(12,16)(13,18)(15,19)(20,27)(21,26)(22,32)(23,33)(24,30)(25,31)(28,35)(29,34)(36,39)(37,38)

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6440);