C₇×#11854 #13189070

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11854

Subgroups

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

412 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 102 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)(10,16,18,23)(11,15)(12,14)(13,17,21,19)(20,22)(24,25,28,33,29,32,27,31)(26,34,35,39,36,30,37,38);(8,10)(9,13)(11,19)(12,18)(14,21)(15,23)(16,22)(17,20)(24,26,29,36)(25,30,32,34)(27,37,28,35)(31,38,33,39);(9,14)(13,21)(15,22)(16,23)(24,27,29,28)(25,31,32,33)(26,35,36,37)(30,38,34,39)

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, 4842);