C₇×#8126 #13185342

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#8126

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

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, 1114);