C₇×#23470 #13200686

order	1792
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇×#23470

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

1178 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 334 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)(8,9,11,13)(10,14,15,12)(16,17)(18,22)(19,23)(20,24)(21,25)(26,36)(27,34)(28,35)(29,37)(30,38)(31,39)(32,40)(33,42)(41,46)(43,45)(44,47);(8,10,11,15)(9,12,13,14)(16,18)(17,22)(19,26)(20,27)(21,28)(23,36)(24,34)(25,35)(29,41)(30,42)(31,43)(32,44)(33,38)(37,46)(39,45)(40,47);(16,19,21,30)(17,23,25,38)(18,26,28,42)(20,31,32,29)(22,33,35,36)(24,39,40,37)(27,43,44,41)(34,46,47,45);(16,20,21,32)(17,24,25,40)(18,27,28,44)(19,29,30,31)(22,34,35,47)(23,37,38,39)(26,41,42,43)(33,45,36,46)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 16458);