C₇×#11838 #13189054

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×#11838

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1514
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₂³×C₇	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 280 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)(10,19)(11,23,25,33)(12,15)(13,18)(14,20,30,38)(16,27,34,37)(17,26)(21,35)(22,36)(24,29,39,31)(28,32);(8,10,13,22)(9,14,18,30)(11,24,25,39)(12,28,26,21)(15,29,17,31)(16,32,34,35)(19,23,36,33)(20,37,38,27);(9,15,18,17)(11,25)(14,29,30,31)(16,33,34,23)(19,35,36,32)(20,38)(24,39)(27,37)

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, 4826);