#72229152

order	13872
automorphism	?
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃×C₁₇² ⋊ C₁₆
Sylow	C₁₇² ⋊ C₃⋊C₁₆
Sylow	C₃ ⋊ #72213346

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₇²	C₁₆
prime radical	C₃×C₁₇⋊D₁₇	C₈
cosocle	C₃×#72213191	C₂
LCS	G ▷ C₃×C₁₇²
UCS	C₁

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 867

A minimal faithful permutation representation — 3 generators on 37 points:

(2,3)(5,6,7,11,8,12,10,17,9,13,14,19,15,16,18,20)(22,23,24,28,25,29,27,34,26,30,31,36,32,33,35,37);(1,2,3)(21,22,27,29,32,30,36,34,24,31,37,28,23,25,33,35,26);(4,5,10,12,15,13,19,17,7,14,20,11,6,8,16,18,9)

Polycyclic presentation:

2^4,3,17^2|a^2=b,b^2=c,c^2=d,e^a=e^2,f^a=f^12,f^b=f^8,f^c=f^13,f^d=f^16,g^a=g^12,g^b=g^8,g^c=g^13,g^d=g^16

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213346, #72259137, #72259142
maximal normal subgroup of	D₃×#72213346, #72259137, #72259142

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 17, 17]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^13);;
gap> SetConjugate(coll, 6, 4, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^13);;
gap> SetConjugate(coll, 7, 4, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);