#72247652

order	2304
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

derived

C₂²⋊(C₂²⋊A₄) ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#87588⋊C₃
commutator, hypocenter	C₂²⋊(C₂²⋊A₄)	C₃×C₄
Fitting, socle	C₂⁷	C₃×D₃
prime radical	C₂×#10596782	C₂
cosocle	C₂×C₂²⋊(C₂²⋊A₄)	C₂×C₃
LCS	G ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂

2114 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 2114 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 16
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,3,4)(5,8,7,6)(9,16,13,15,11,14)(10,12);(6,7,8)(11,13,12)(14,15,16)

Polycyclic presentation:

2,3,2,3,2^6|a^2=c,d^a=d^2,e^a=h,e^b=ef,e^d=eh,f^a=g,f^b=e,f^d=fg,g^a=f,g^b=h,g^d=f,h^a=e,h^b=gh,h^d=e,i^a=ij,i^d=j,j^d=ij

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247682, C₄×#87588⋊C₃, #72248184, …
maximal normal subgroup of	C₄×#87588⋊C₃, #72248184, #72253634
maximal quotient of	#72247720

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