#72228182

order	2888
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁₉

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow

C₁₉² ⋊ Q₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, prime radical, cosocle	C₁₉⋊D₁₉	C₂²
Fitting, socle, hypocenter	C₁₉²	Q₈
LCS	G ▷ C₁₉⋊D₁₉ ▷ C₁₉²
UCS	C₁

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 2 generators on 76 points:

(1,25,63,11)(2,21,71,29)(3,4,57,52)(5,10,68,47)(6,62,65,19)(7,37,72,16)(8,49,33,18)(9,12,32,22)(13,53,64,31)(14,46,39,38)(15,69,56,35)(17,36,45,60)(20,40,67,42)(23,70,73,48)(24,43,55,76)(26,30,54,66)(27,34)(28,59,61,75)(41,44)(50,74,58,51);(1,2)(3,5)(4,10,12,6)(7,18,17,13)(8,21,23,15)(9,26,27,19)(11,16,31,29)(14,25,36,24)(20,43,45,64)(22,38,48,47)(28,56,57,44)(30,59,41,34)(32,62,58,39)(33,42,53,60)(35,51,65,52)(37,50,54,55)(40,61,68,73)(46,70,71,63)(49,69,74,72)(66,75,67,76)

Polycyclic presentation:

2^3,19^2|a^2=c,b^2=c,b^a=bc,d^a=d^18e^6,d^b=d^5e^2,d^c=d^18,e^a=d^6e,e^b=d^6e^14,e^c=e^18

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72228183
maximal subgroup of	#72228183
maximal normal subgroup of	#72228183

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 19, 19]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^18*F.5^6);;
gap> SetConjugate(coll, 4, 2, F.4^5*F.5^2);;
gap> SetConjugate(coll, 4, 3, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.4^6*F.5);;
gap> SetConjugate(coll, 5, 2, F.4^6*F.5^14);;
gap> SetConjugate(coll, 5, 3, F.5^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);