#72247657

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	#1020⋊A₄
commutator, hypocenter	C₂²×C₂²⋊A₄	C₃×C₄
Fitting, socle	C₂⁷	C₃×D₃
prime radical	C₂×#10596782	C₂
cosocle	C₂³×C₂²⋊A₄	C₂×C₃
LCS	G ▷ C₂²×C₂²⋊A₄
UCS	C₁ ◁ C₂

2522 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 2522 conjugacy classes of subgroups — too many to draw. Only its 19 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)(6,8,7)(9,16,13,15,11,14)(10,12);(5,8)(6,7)(11,13,12)(14,15,16)

Polycyclic presentation:

2,3,2^3,3,2^4|a^2=c,d^b=e,e^b=de,f^a=f^2,g^b=ghi,g^f=j,h^a=i,h^b=gj,h^f=hi,i^a=h,i^b=j,i^f=h,j^a=gj,j^b=ij,j^f=gj

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247720, C₄×#1020⋊A₄, #72248196, …
maximal normal subgroup of	C₄×#1020⋊A₄, #72248196, #72253634
maximal quotient of	#72247682

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