#72228563

order	24200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₅×C₁₁² ⋊ C₂×C₄×C₅
Hall	C₅×#10597243 ⋊ C₂×C₄
Sylow	C₁₁² ⋊ C₂×C₅×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₁₁²	C₂×C₄×C₅
prime radical	D₅×#11925892	C₂
cosocle	D₅×C₁₁²	C₂²×C₅
LCS	G ▷ C₅×C₁₁²
UCS	C₁

256 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 27
transitive degree	μ_t(G) = 110

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2^2,5,2,5,11^2|a^2=d,e^a=e^3,e^d=e^4,f^a=f^10,f^b=f^10,f^c=f^3,g^a=f^2g,g^b=g^10,g^c=g^3

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11925892⋊C₂
maximal normal subgroup of	AGL(1,𝔽₅)×#11925892⋊C₂

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