C₇×#10648 #13187864

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#10648

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

236 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 82 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)(8,9,12,15,13,17,11,16)(10,14,18,23,20,21,19,22)(24,25)(26,34,37,47)(27,31)(28,32)(29,33)(30,36,44,41)(35,50,38,49)(39,45)(40,46)(42,48)(43,52,54,51)(53,55);(8,10)(9,14)(11,19)(12,18)(13,20)(15,23)(16,22)(17,21)(24,26,29,37)(25,30,33,44)(27,38,40,35)(28,41,42,36)(31,43,46,54)(32,47,48,34)(39,51,53,52)(45,50,55,49);(9,15)(10,18)(11,12)(14,21)(16,17)(19,20)(24,27,28,39,29,40,42,53)(25,31,32,45,33,46,48,55)(26,35,36,51,37,38,41,52)(30,43,34,49,44,54,47,50)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3636);