C₇×#9334 #13186550

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9334

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

398 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,9,12,16,13,15,22,10)(11,18,20,23,19,14,21,17)(24,25,29,33,30,35,28,34)(26,36,39,46,40,49,38,31)(27,41,43,32,44,47,42,48)(37,45,52,54,53,55,51,50);(8,10,12,15,13,16,22,9)(11,17,21,18,19,23,20,14)(24,26)(25,31,35,46)(27,37)(28,38)(29,39)(30,40)(32,45,48,55)(33,36,34,49)(41,50,47,54)(42,51)(43,52)(44,53);(8,11,13,19)(9,14,15,18)(10,17,16,23)(12,21,22,20)(24,27,30,44)(25,32,35,48)(26,37,40,53)(28,43,29,42)(31,45,46,55)(33,41,34,47)(36,50,49,54)(38,52,39,51)

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, 2322);