C₇×#7239 #13184455

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7239

Subgroups

	subgroup	G/H
center	C₄×C₇	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₇
Frattini, cosocle	#405	C₂²×C₇
socle	C₂×C₇	#1290
prime radical	C₂³×C₇	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ G

148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 71 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 227);