C₇×#20691 #13197907

order	1792
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇×#20691

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇×Q₈ ◁ G

1266 conjugacy classes of subgroups; only the 446 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 446 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,22)(19,29,31,23)(20,24)(21,25)(26,40,42,34)(27,35)(28,36)(30,43,45,37)(32,38)(33,39)(41,47,44,46);(9,12)(13,15)(16,18,21,28)(17,22,25,36)(19,26,31,42)(20,32,33,27)(23,34,29,40)(24,38,39,35)(30,44,45,41)(37,47,43,46);(8,10,11,14)(9,13,12,15)(16,19)(17,23)(18,26)(20,30)(21,31)(22,34)(24,37)(25,29)(27,41)(28,42)(32,44)(33,45)(35,46)(36,40)(38,47)(39,43);(16,20,21,33)(17,24,25,39)(18,27,28,32)(19,30,31,45)(22,35,36,38)(23,37,29,43)(26,41,42,44)(34,46,40,47)

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 13679);