#72258803

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×C₃×Q₈ ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×S₄
commutator, cosocle	C₂×C₃×SL(2,𝔽₃)	C₂⁴
Fitting	C₂³×C₃×Q₈	C₂×D₃
socle	C₂²×C₃	C₂³×S₄
hypercenter	C₂⁴	D₃×S₄
hypocenter	C₃×SL(2,𝔽₃)	C₂²×D₄
LCS	G ▷ C₂×C₃×SL(2,𝔽₃) ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂⁴

3773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(8,10,18)(9,20,23,13,14,22)(12,19,17)(15,21);(2,3)(10,11)(14,15)(16,19)(17,18)(20,21)(22,23);(5,7);(4,5,6,7)(8,9)(10,15)(11,14)(12,13)(16,20)(17,23)(18,22)(19,21)

Polycyclic presentation:

2^5,3^2,2^3|a^2=e,h^2=j,i^2=j,b^a=be,d^a=dj,f^a=f^2,f^c=f^2,g^a=f^2g,g^c=g^2,h^a=i,h^c=i,h^f=ij,h^g=hij,i^a=h,i^c=h,i^f=hi,i^g=hj,i^h=ij

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

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