C₂×C₃²×M₆₄ #11769306

order	1152
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₁₆ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₂×C₃²×M₆₄

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₁₆	C₂²
commutator	C₂	C₂²×C₃²×C₁₆
Frattini, cosocle	C₁₆	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₁₆
prime radical	C₂³×C₃²	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₁₆ ◁ G

336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 40 points:

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

Conjugacy classes & character table

720 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×C₃⋊D₃×M₆₄, C₂×#11772672
maximal subgroup of	C₂×C₃²×#1398, C₂×C₃⋊D₃×M₆₄, C₂×#11772672
maximal normal subgroup of	C₂×C₃²×#1398, C₂×C₃⋊D₃×M₆₄, C₂×#11772672

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3317);

C2×C32×M64 #11769306

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×C₃²×M₆₄ #11769306