C₂³×C₇×#126 #13239878

order	1792
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₇ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂³×C₇×#126

Subgroups

	subgroup	G/H
center, socle	C₂⁵×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
prime radical	C₂⁶×C₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₇ ◁ G

9486 conjugacy classes of subgroups; only the 6366 normal subgroups are shown.

normal subgroups only

This group has 9486 conjugacy classes of subgroups — too many to draw. Only its 6366 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) = 29
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 6 generators on 29 points:

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

Conjugacy classes & character table

784 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 55650);