C₇×#11916 #13189132

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11916

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1533
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#97	C₂³×C₇
prime radical	C₂³×C₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×M₁₆ ◁ G

274 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 224

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4904);