C₇×#19711 #13196927

order	1792
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#19711

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂³×C₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

636 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 55 points:

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

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, 12699);