C₇×#9985 #13187201

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9985

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

502 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 448

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2973);