C₂²×C₉×#126 #11768159

order	1152
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₉ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂²×C₉×#126

Subgroups

	subgroup	G/H
center	C₂⁴×C₉	C₂³
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂⁴×C₃	C₂³×C₃
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

1980 conjugacy classes of subgroups; only the 1308 normal subgroups are shown.

normal subgroups only

This group has 1980 conjugacy classes of subgroups — too many to draw. Only its 1308 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) = 576

A minimal faithful permutation representation — 5 generators on 29 points:

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

Conjugacy classes & character table

504 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2170);