#72227634

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×C₂²⋊C₄ ⋊ C₄
Sylow	C₅² ⋊ #2342

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1043
commutator, socle	C₂²×C₅²	C₂³×C₄
Fitting	C₂×C₅²×C₂²⋊C₄	C₄
prime radical	C₂⁴×C₅⋊D₅	C₂²
cosocle	C₂²×C₅⋊D₅	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#1043
hypocenter	C₅²	#2342
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

3168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 4 generators on 22 points:

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

Polycyclic presentation:

2^7,5^2|a^2=g,d^2=e,b^a=bf,d^c=dg,h^d=h^3,h^e=h^4,i^d=i^3,i^e=i^4

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258310
maximal normal subgroup of	#72258310

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