C₇×#21828 #13199044

order	1792
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#21828

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 ◁ C₂²×C₇×C₄⋊C₄ ◁ G

990 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 366 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) = 448

A minimal faithful permutation representation — 4 generators on 39 points:

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

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, 14816);