#72303813

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#410
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂²	C₂×#1799
prime radical	C₂²×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂²×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

5495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=fghij,b^a=be,c^a=cef,c^b=cefgh,d^b=di,d^c=dj,e^b=ei,f^b=fi,g^a=gi,g^c=gij,g^d=gj,g^e=gj,g^f=gj,h^a=hi,h^b=hi,h^c=hij,h^d=hj,h^e=hj,h^f=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303815, #72313152, #72313201, …
maximal normal subgroup of	#72303815, #72313152, #72313201, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);