C₇×#19056 #13196272

order	1792
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#19056

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3549	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

992 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 222 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)(10,18)(11,15)(12,16)(13,17)(14,20)(19,22)(21,23)(24,25,29,32)(26,33,35,38)(27,31,28,36)(30,37,39,34);(8,10)(9,14)(11,19)(12,20)(13,21)(15,18)(16,23)(17,22)(24,26)(25,30)(27,34)(28,37)(29,35)(31,38)(32,39)(33,36);(8,10)(9,14)(11,20)(12,19)(13,21)(15,23)(16,18)(17,22)(27,28)(31,36)(33,38)(34,37);(8,11,13,12)(9,15,17,16)(10,19,21,20)(14,18,22,23)(24,27,29,28)(25,31,32,36)(26,34,35,37)(30,38,39,33)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 12044);