C₇×#5428 #13057148

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

solvable

Aliases

direct product

C₇×#5428

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
prime radical	C₇×#994	C₂
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

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

normal subgroups only

This group has 138 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) = 55
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 2 generators on 55 points:

(1,2,3,4,5,6,7)(8,26,53,13,29,36,15,32,43,10,44,28)(9,18,48,34,37,42,20,39,45,16,33,52)(11,12,50,22,31,46,14,30,51,25,23,54)(17,38,49,21,40,47,24,35,27,19,41,55);(8,9,15,20)(10,21,13,19)(11,24,14,17)(12,27,30,49)(16,25,34,22)(18,36,39,28)(23,42,31,52)(26,45,32,48)(29,47,44,55)(33,50,37,51)(35,46,38,54)(40,43,41,53)

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, 6437);