#72233470

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting

#63103 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	C₂²×#1020
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini, socle	C₂	C₂³×#1020
Fitting	#63103	D₃
hypercenter	#41	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₄ ◁ #41

12655 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 12655 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^4,3,2^5|c^2=j,c^b=cj,d^b=dj,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^a=i,h^e=hi,h^f=hj,i^a=h,i^e=h,i^g=ij

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233475, #72240983, #72240994, …
maximal normal subgroup of	#72240983, #72240994, #72240998, …
maximal quotient of	C₄×#11715839, C₂×#72233470, #72341757

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