C₇×#9422 #13186638

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9422

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 384 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)(8,9,14,17,13,10,12,16)(11,15,21,23,20,18,19,22)(24,25)(26,35)(27,39,42,49)(28,43)(29,33)(30,34)(31,37)(32,41,48,40)(36,51,53,45)(38,46)(44,47)(50,52,55,54);(8,10,13,9)(11,18,20,15)(12,16,14,17)(19,22,21,23)(24,26,28,37,30,38,29,44)(25,31,33,35,34,47,43,46)(27,36,41,54,42,53,40,52)(32,45,49,50,48,51,39,55);(8,11,13,20)(9,15,10,18)(12,21,14,19)(16,23,17,22)(24,27)(25,32)(26,36)(28,41)(29,40)(30,42)(31,45)(33,49)(34,48)(35,50)(37,54)(38,53)(39,43)(44,52)(46,55)(47,51)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2410);