Dic₇₇₂₂ #72272151

order	30888
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₇₂₂

Splittings

Hall	C₁₁×C₁₃×C₂₇ ⋊ Q₈
Hall	C₁₃×C₂₇ ⋊ Dic₂₂
Hall	C₁₁×C₂₇ ⋊ Dic₂₆
Hall	C₁₁×C₁₃ ⋊ Dic₅₄
Sylow	C₂₇ ⋊ Dic₂₈₆
Sylow	C₁₃ ⋊ Dic₅₉₄
Sylow	C₁₁ ⋊ Dic₇₀₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₈₆₁
commutator, cosocle	C₂×C₁₁×C₁₃×C₂₇	C₂²
Frattini	C₂×C₉	C₂×D₄₂₉
Fitting	C₄×C₁₁×C₁₃×C₂₇	C₂
socle, prime radical	C₂×C₃×C₁₁×C₁₃	C₂×D₉
hypercenter	C₄	D₃₈₆₁
hypocenter	C₁₁×C₁₃×C₂₇	Q₈
LCS	G ▷ C₂×C₁₁×C₁₃×C₂₇ ▷ C₁₁×C₁₃×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 30888

A minimal faithful permutation representation — 2 generators on 59 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35)(36,37,38,39,40,41,42,43,44,45,46)(47,48,49,50,51,52,53,54,55,56,57,58,59);(1,5,3,7)(2,6,4,8)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,46)(38,45)(39,44)(40,43)(41,42)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2^3,3,11,13,3^2|a^2=c,b^2=c,d^3=g,g^3=h,b^a=bc,d^a=d^2g^2h^2,e^a=e^10,f^a=f^12,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

7725 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₄₄₄
maximal normal subgroup of	Dic₁₅₄₄₄

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 11, 13, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7722 #72272151