C₇×#11836 #13189052

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#11836

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1514
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₇×#469	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 286 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)(10,20,22,37)(11,25,27,32)(12,18)(13,16)(14,19)(15,21)(17,28,34,29)(23,33,24,36)(26,35)(30,39)(31,38);(8,10)(9,15)(11,26)(12,24)(13,23)(14,22)(16,30)(17,33)(18,35)(19,31)(20,25)(21,29)(27,39)(28,38)(32,37)(34,36);(9,16,19,18)(11,27)(15,30,31,35)(17,25,34,32)(20,36,37,33)(21,38)(26,39)(28,29)

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, 4824);