#72258634

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting

#62852 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×GL(2,ℤ/4)
commutator, cosocle	C₂×#41⋊C₃	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	#62852	D₃
hypercenter	#536	S₄
hypocenter	SL(2,𝔽₃)	C₂×#127
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ #117 ◁ #536

3972 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:

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

Polycyclic presentation:

2^6,3,2^3|a^2=e,c^2=f,f^2=j,h^2=j,i^2=j,b^a=be,c^b=cf,d^a=dj,f^b=fj,g^b=g^2,g^c=g^2,h^b=ij,h^c=ij,h^g=i,i^b=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	#72258787
maximal normal subgroup of	#72258787

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, 1, F.5);;
gap> SetPower(coll, 3, F.6);;
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, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 2, 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);