C₇×#12338 #13189554

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12338

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3403	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#446 ◁ G

320 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 78 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)(20,32,35,43)(23,38,40,29)(25,31,44,34)(28,46,37,42)(33,39,47,41);(10,14)(12,15)(16,18)(17,23)(19,28)(20,33)(21,37)(22,30)(24,39)(25,42)(26,40)(27,32)(29,34)(31,38)(35,47)(36,41)(43,45)(44,46);(8,9,11,13)(10,15,14,12)(16,19,22,21)(17,24,26,36)(18,28,30,37)(20,34,35,31)(23,39,40,41)(25,43,44,32)(27,42,45,46)(29,47,38,33)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5326);