#72240983

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Splittings

Fitting

#10574874 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	C₂²×#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁵
Frattini	C₂³	C₂⁴×S₄
Fitting	#10574874	D₃
socle	C₂	C₂³×#1020⋊C₂
hypercenter	#41	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₄ ◁ #41

At least 8114 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^5,3,2^5|b^2=k,e^2=k,g^2=k,h^2=k,b^a=bk,c^b=ck,e^b=ek,f^d=f^2,g^a=gik,g^d=gh,g^f=h,h^a=hijk,h^d=hk,h^f=ghk,h^g=hk,i^d=j,i^f=ij,i^h=ik,j^d=i,j^f=i,j^g=jk,j^h=jk

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245616, #72342856, C₂×#72240983
maximal normal subgroup of	#72245616, #72342856, C₂×#72240983
maximal quotient of	C₄×#72217027, #72341901, C₂×#72240983
automorphism group of	#3407

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);