C₉×#1381 #11766102

order	1152
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1381

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉	D₈
commutator	C₈	C₄²×C₉
Frattini, cosocle	C₂²×C₃×C₈	C₂²×C₃
socle	C₂²×C₃	C₃×#107
prime radical	C₂³×C₃	C₃×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

192 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 45
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,18,23,20,26,32,25)(16,27,29,41,31,42,43,36)(17,24,19,22,33,39,34,40)(21,30,35,44,38,45,37,28);(11,13)(14,16,19,30,18,29,33,44,20,31,34,45,32,43,17,28)(15,21,24,36,25,37,40,42,26,38,39,41,23,35,22,27)

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 113);