C₇×#5429 #13057149

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

solvable

Aliases

direct product

C₇×#5429

Splittings

Fitting

C₇×H(5,𝔽₂) ⋊ 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×H(5,𝔽₂)	D₃
hypercenter	C₇×Q₈	S₄
hypocenter	SL(2,𝔽₃)	C₇×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×Q₈

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

normal subgroups only

This group has 156 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) = 31
transitive degree	μ_t(G) = 168

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

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₇×SO⁺(4,𝔽₃), ΓL(1,𝔽₈)×#5429, D₇×#5429
maximal normal subgroup of	ΓL(1,𝔽₈)×#5429, D₇×#5429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6438);