#72226324

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

D₄×H(5,𝔽₂) ⋊ 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	D₄×H(5,𝔽₂)	D₃
hypercenter	D₄×Q₈	S₄
hypocenter	SL(2,𝔽₃)	D₄²
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ D₄×Q₈

3526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226331, #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, 4, 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, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*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);