C₇×#12333 #13189549

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12333

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×Q₈ ◁ G

324 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 896

A minimal faithful permutation representation — 3 generators on 47 points:

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

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, 5321);