C₇×#12027 #13189243

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12027

Subgroups

	subgroup	G/H
center	C₄×C₇	#427
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂×C₇	#1568
prime radical	C₂³×C₇	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₇×#97 ◁ C₇×#405 ◁ G

180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,9,13,20,15,21,39,52,16,22,38,19,14,42,12,37)(10,23,26,44,28,54,25,45,29,50,40,68,27,62,60,17)(11,30,33,64,35,49,32,51,36,65,41,69,34,18,46,48)(24,55,57,43,59,70,67,71,31,66,61,47,58,53,56,63);(8,10,12,25,14,27,38,26,16,29,39,60,15,28,13,40)(9,17,19,23,21,44,37,54,22,45,20,50,42,68,52,62)(11,31,32,67,34,59,33,57,36,24,46,56,35,58,41,61)(18,47,48,70,30,63,64,66,49,43,51,53,65,71,69,55);(8,11,14,34,16,36,15,35)(9,18,21,30,22,49,42,65)(10,24,27,58,29,31,28,59)(12,32,38,33,39,46,13,41)(17,43,44,71,45,47,68,63)(19,48,37,64,20,51,52,69)(23,53,54,55,50,70,62,66)(25,56,26,61,60,67,40,57)

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, 5015);