C₇×#12170 #13189386

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12170

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 356 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) = 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)(10,19)(11,14)(12,27)(13,18)(15,21)(16,23)(17,25)(20,33)(22,32)(24,28)(26,29)(30,39)(31,38)(34,36)(35,37);(9,14)(11,23)(12,25)(15,28)(16,17)(18,29)(19,33)(20,36)(22,37)(24,38)(26,39)(27,30)(31,32)(34,35);(8,10,13,21)(9,15)(11,24,26,20)(12,22,25,37)(14,28)(16,31)(17,32)(18,19)(23,38,39,36)(27,35)(29,33)(30,34)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5158);