C₇×#10614 #13187830

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#10614

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 276 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) = 39
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,14,17,13,18,12,22)(10,19,21,23,11,16,20,15)(24,25)(26,28)(27,35)(29,32)(30,33)(31,37)(34,38)(36,39);(8,10,14,21,13,11,12,20)(9,15,22,16,18,23,17,19)(24,26,29,33,30,32,28,25)(27,34,37,31,38,35,36,39);(8,11,13,10)(9,16,18,19)(12,20,14,21)(15,17,23,22)(24,27,30,38)(25,31,33,39)(26,34,32,35)(28,37,29,36)

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, 3602);