C₇×#8792 #13186008

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₇×#8792

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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)(10,15)(11,20)(12,22)(13,17)(14,18)(16,21)(19,23)(24,25)(26,34,37,30)(27,38,40,53)(28,33)(29,32)(31,39,44,54)(35,48)(36,47,51,43)(41,45)(42,49)(46,55)(50,52);(8,10,12,18,13,19,21,11)(9,14,16,15,17,20,22,23)(24,26,28,36,29,37,41,51)(25,30,33,43,32,34,45,47)(27,35,39,49,40,50,54,55)(31,42,38,52,44,46,53,48);(8,11,21,19,13,18,12,10)(9,15,22,14,17,23,16,20)(24,27,29,40)(25,31)(26,35,37,50)(28,39,41,54)(30,42)(32,44)(33,38)(34,46)(36,49,51,55)(43,52)(45,53)(47,48)

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, 1780);