C₇×#14336 #13191552

order	1792
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₇×#14336

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

3998 conjugacy classes of subgroups; only the 702 normal subgroups are shown.

normal subgroups only

This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 702 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) = 35
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 7324);