#72229162

order	8112
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₁₃² ⋊ C₃×C₄
Hall	#72229144 ⋊ C₃
Hall	#92788 ⋊ C₄⋊C₄
Sylow	C₁₃² ⋊ C₃×C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#14523144
commutator, socle	C₂×C₁₃²	C₂×C₃×C₄
Fitting	C₄×C₁₃²	C₃×C₄
prime radical	C₂×#11759815	C₂²
cosocle	C₂×C₁₃⋊D₁₃	C₂²×C₃
hypercenter	C₄	#14523144
hypocenter	C₁₃²	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄

416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 676

A minimal faithful permutation representation — 3 generators on 30 points:

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

Polycyclic presentation:

2^2,3,2^2,13^2|a^2=d,b^2=e,b^a=be,f^a=f^8,f^c=f^3,f^d=f^12,g^a=g^8,g^c=g^3,g^d=g^12

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#14523144, C₂×#72229162, #72241938, …
maximal normal subgroup of	D₄×#14523144, C₂×#72229162, #72241938, …
maximal quotient of	C₂×#72229162

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