#72244658

order	12096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₇ ⋊ C₃×D₃²
derived	C₇×A₄² ⋊ C₂²×C₃
Sylow	C₇ ⋊ C₃×S₄²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×A₄²	C₂²×C₃
Fitting, socle	C₂⁴×C₇	C₃×D₃²
LCS	G ▷ C₇×A₄²
UCS	C₁

1202 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 15
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2^2,3^3,7,2^4|d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=f^6,f^c=f^4,g^d=h,g^e=gh,h^a=gh,h^b=gh,h^d=gh,h^e=g,i^a=gi,i^b=gi,i^d=gi,i^e=j,j^b=hij,j^d=ghj,j^e=ij

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄²×AGL(1,𝔽₇)
maximal normal subgroup of	S₄²×AGL(1,𝔽₇)

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