C₇×#21796 #13199012

order	1792
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₇×#21796

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂×C₇×#537	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₇×C₄⋊C₄ ◁ G

1334 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1334 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 31 points:

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

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, 14784);