C₇×#5389 #13057110

order	1344
isoclinism	isoclinic class of #5386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

solvable

Aliases

direct product

C₇×#5389

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂²×C₇
Frattini	C₂²	C₂×C₇×S₄
Fitting	C₄×C₇×Q₈	D₃
prime radical	C₂×C₇×SL(2,𝔽₃)	C₂²
hypercenter	C₂×C₄×C₇	S₄
hypocenter	SL(2,𝔽₃)	C₇×Q₈
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇

112 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 31 points:

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

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, 6399);