C₇×#9006 #13186222

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9006

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#120
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#386
prime radical	C₂³×C₇	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

376 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 896

A minimal faithful permutation representation — 3 generators on 55 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1994);