C₇×#13560 #13190776

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13560

Subgroups

	subgroup	G/H
center	C₄×C₇	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₇
Frattini, cosocle	#127	C₂³×C₇
socle	C₂×C₇	C₂×AΣL(1,𝔽₁₆)
prime radical	C₇×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

486 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 90 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)(8,9)(10,19,21,14)(11,15)(12,17,26,16)(13,18)(20,33,34,27)(22,32,25,28)(23,30,37,29)(24,31)(35,39,36,38);(8,10)(9,14)(11,20)(12,25)(13,22)(15,27)(16,19)(17,32)(18,28)(21,26)(23,36)(24,35)(29,33)(30,39)(31,38)(34,37);(8,11,13,24)(9,15,18,31)(10,20,22,35)(12,23,26,37)(14,27,28,38)(16,29,17,30)(19,33,32,39)(21,34,25,36)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6548);