C₇×#8796 #13186012

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#8796

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 228 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)(8,9,12,18,14,19,13,17)(10,16,11,15,21,22,20,23)(24,25,29,35,30,33,28,34)(26,36,38,48,39,31,43,46)(27,40,41,47,42,49,44,32)(37,51,53,50,54,55,52,45);(8,10,13,20,14,21,12,11)(9,15,18,22,19,23,17,16)(24,26,28,38,30,39,29,43)(25,31,34,46,33,36,35,48)(27,37,41,52,42,54,44,53)(32,45,40,55,47,50,49,51);(8,11,12,21,14,20,13,10)(9,16,17,23,19,22,18,15)(24,27)(25,32,33,47)(26,37)(28,41)(29,44)(30,42)(31,45,36,50)(34,40,35,49)(38,52)(39,54)(43,53)(46,55,48,51)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1784);