#72229249

order	5408
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

derived, Sylow

C₁₃² ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#261²
commutator, hypocenter	C₁₃²	C₄×C₈
Fitting, socle	C₂×C₁₃²	C₄²
prime radical	C₂×C₁₃⋊D₁₃	C₂×C₄
cosocle	#13062577	C₂²
LCS	G ▷ C₁₃²
UCS	C₁ ◁ C₂

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 2 generators on 34 points:

(10,11,12,14)(13,16,17,21)(15,19,20,18)(22,23,29,25)(24,32,34,31)(26,33,30,27);(1,2,3,5,4,6,7,8)(9,10,13,16,20,14,18,19,11,15,21,17,12)(23,25,24,27)(26,31,28,34)(29,30,33,32)

Polycyclic presentation:

2^5,13^2|a^2=c,b^2=d,c^2=e,f^a=f^8,f^b=f^8,f^c=f^12,f^d=f^12,g^a=f^6g,g^b=g^8,g^c=f^2g,g^d=g^12

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72229256, #72229293
maximal normal subgroup of	#72229256, #72229293

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, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 6, 1, F.6^8);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> SetConjugate(coll, 6, 4, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.6^6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);