C₇×#11990 #13189206

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11990

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1551
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#405 ◁ G

268 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,12,18,13,17,31,43,14,20,32,46,34,19,47,45)(10,21,23,49,24,38,53,63,25,37,54,44,33,15,30,36)(11,26,28,55,29,16,39,40,27,42,56,68,57,41,65,67)(22,35,51,61,52,48,69,70,50,62,64,71,59,60,58,66);(8,10,14,25)(9,15,20,38)(11,22,27,50)(12,30,32,53)(13,33,34,24)(16,35,41,62)(17,37,19,21)(18,44,46,49)(23,31,54,47)(26,48,42,60)(28,58,56,69)(29,59,57,52)(36,43,63,45)(39,64,65,51)(40,66,67,70)(55,61,68,71);(8,11)(9,16,20,41)(10,22)(12,28)(13,29)(14,27)(15,35,38,62)(17,42,19,26)(18,40,46,67)(21,48,37,60)(23,51)(24,52)(25,50)(30,58)(31,39)(32,56)(33,59)(34,57)(36,61,63,71)(43,68,45,55)(44,66,49,70)(47,65)(53,69)(54,64)

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, 4978);