C₇×#11576 #13188792

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11576

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

426 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4564);