#72258705

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×S₄
commutator	C₂×SL(2,𝔽₃)	C₂³×C₄
Fitting	#62811	D₃
socle	C₂²	C₂³×SL(2,ℤ/4)
prime radical	#62811⋊C₃	C₂
cosocle	C₂×#41⋊C₃	C₂⁴
hypercenter	#525	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₄×D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄ ◁ #525

3575 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)(6,35,52,13,32,50)(7,25,43)(8,26,22)(9,42,19)(10,39,41)(11,24,40)(12,46,51,27,36,47)(14,15,33,30,31,48)(16,29,49,28,45,34)(17,18,23)(20,37,38);(5,7,8,17,11,20,19,10)(6,12,14,28,13,27,30,16)(9,22,21,40,26,42,24,44)(15,33,32,50,31,48,35,52)(18,38,37,41,39,43,25,23)(29,47,46,49,45,51,36,34);(2,4);(1,2,3,4)(5,6)(7,12)(8,14)(9,15)(10,16)(11,13)(17,28)(18,29)(19,30)(20,27)(21,32)(22,33)(23,34)(24,35)(25,36)(26,31)(37,46)(38,47)(39,45)(40,50)(41,49)(42,48)(43,51)(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,d^a=dj,g^c=g^2,h^c=ij,h^g=i,i^c=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

124 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, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
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);