#72258683

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#3585 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×S₄
commutator, cosocle	C₂×#41⋊C₃	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	C₂×#3585	D₃
hypercenter	C₂×#137	S₄
hypocenter	SL(2,𝔽₃)	C₂×#120
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

5100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 52 points:

(5,40,50)(6,32,52,13,31,36)(7,25,43)(8,9,49)(10,39,47)(11,22,37)(12,51,23,16,34,44)(14,15,45,29,30,48)(17,28,46,27,33,41)(18,19,24)(20,35,42)(21,26,38);(7,18)(8,21)(9,24)(10,20)(12,17)(14,29)(15,23)(16,27)(19,37)(22,42)(25,49)(26,47)(28,45)(30,44)(31,46)(32,41)(33,48)(34,36)(35,38)(39,50)(40,43)(51,52);(2,4)(5,7,11,20)(6,12,13,16)(8,18,21,10)(9,19,26,39)(14,17,29,27)(15,28,30,33)(22,35,40,25)(23,36,44,52)(24,38,47,49)(31,34,32,51)(37,42,50,43)(41,45,46,48);(1,2)(3,4)(5,6)(7,12)(8,14)(9,23)(10,27)(11,13)(15,24)(16,20)(17,18)(19,36)(21,29)(22,41)(25,48)(26,44)(28,38)(30,47)(31,43)(32,42)(33,49)(34,37)(35,45)(39,52)(40,46)(50,51)

Polycyclic presentation:

2^6,3,2^3|b^2=j,f^2=j,h^2=j,i^2=j,b^a=be,c^b=cf,d^a=dj,f^b=fj,f^c=fj,g^a=g^2,g^c=g^2,h^a=ij,h^c=ij,h^g=i,i^a=hj,i^c=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258773
maximal normal subgroup of	#72258773

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, 3, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);