D₇₇₄₉ #72212902

order	15498
automorphism	Aut(G) order 33475680
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₇₇₄₉

Splittings

Fitting, derived, Hall	C₇×C₂₇×C₄₁ ⋊ C₂
Hall	C₂₇×C₄₁ ⋊ D₇
Hall	C₇×C₄₁ ⋊ D₂₇
Hall	C₇×C₂₇ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₁₈₉
Sylow	C₂₇ ⋊ D₂₈₇
Sylow	C₇ ⋊ D₁₁₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₇×C₄₁	C₂
Frattini	C₉	D₈₆₁
socle	C₃×C₇×C₄₁	D₉
LCS	G ▷ C₇×C₂₇×C₄₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 7749

A minimal faithful permutation representation — 2 generators on 75 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,34)(30,33)(31,32)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,3,7,41,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^6,d^a=d^40,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

3876 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₇₇₄₉, D₂₃₂₄₇
maximal normal subgroup of	C₂×D₇₇₄₉
maximal quotient of	Dic₇₇₄₉, C₂×D₇₇₄₉, D₂₃₂₄₇

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");;
gap> coll := SingleCollector(F, [2, 3, 7, 41, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7749 #72212902