C₇×#7312 #13184528

order	1792
nilpotency class	4
isoclinism	isoclinic class of #7310
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#7312

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1304
commutator	C₂⁴	C₄²×C₇
Frattini, cosocle	C₂²×#41	C₂²×C₇
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₇×#41 ◁ G

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 39 points:

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

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, 300);