C₇×#7115 #13184331

order	1792
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#7115

Subgroups

	subgroup	G/H
center	C₄×C₇	#329
commutator	C₄²	C₂×C₇×C₈
Frattini, cosocle	#376	C₂²×C₇
socle	C₂×C₇	#1270
prime radical	C₇×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#376 ◁ G

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

normal subgroups only

This group has 248 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 — 2 generators on 39 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 103);