C₇×#8130 #13185346

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#8130

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#95
commutator	C₂³	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂⁴×C₇	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄²×C₇ ◁ G

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 448

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

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

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, 1118);