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C₉²×M₁₆ #13046876

order	1296
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉² · Schur multiplier (H₂) = C₉

Aliases

C₉²×M₁₆

Subgroups

	subgroup	G/H
center	C₄×C₉²	C₂²
commutator	C₂	C₂×C₄×C₉²
Frattini, cosocle	C₃²×C₄	C₂²×C₃²
socle	C₂×C₃²	C₂×C₃²×C₄
prime radical	C₂²×C₃²	C₃²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉² ◁ G

230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

810 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	M₁₆×C₉⋊D₉, #72244240
maximal subgroup of	M₁₆×C₉⋊D₉, C₉²×AGL(1,ℤ/8), #72244240, …
maximal normal subgroup of	M₁₆×C₉⋊D₉, C₉²×AGL(1,ℤ/8), #72244240, …
maximal quotient of	C₉²×#97, C₂×C₉²×M₁₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 594);