D₁₅₄₇₇ #44526244

order	30954
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₄₇₇
unambiguous semidirect product	C₃⋊D₅₁₅₉
unambiguous semidirect product	C₇⋊D₂₂₁₁
unambiguous semidirect product	C₁₁⋊D₁₄₀₇
unambiguous semidirect product	C₆₇⋊D₂₃₁

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₁×C₆₇ ⋊ C₂
Hall	C₇×C₁₁×C₆₇ ⋊ D₃
Hall	C₃×C₁₁×C₆₇ ⋊ D₇
Hall	C₃×C₇×C₆₇ ⋊ D₁₁
Hall	C₁₁×C₆₇ ⋊ D₂₁
Hall	C₇×C₆₇ ⋊ D₃₃
Hall	C₃×C₇×C₁₁ ⋊ D₆₇
Hall	C₃×C₆₇ ⋊ D₇₇
Hall	C₇×C₁₁ ⋊ D₂₀₁
Sylow	C₆₇ ⋊ D₂₃₁
Hall	C₃×C₁₁ ⋊ D₄₆₉
Hall	C₃×C₇ ⋊ D₇₃₇
Sylow	C₁₁ ⋊ D₁₄₀₇
Sylow	C₇ ⋊ D₂₂₁₁
Sylow	C₃ ⋊ D₅₁₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₁×C₆₇	C₂
LCS	G ▷ C₃×C₇×C₁₁×C₆₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 15477

A minimal faithful permutation representation — 2 generators on 88 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88);(2,3)(5,10)(6,9)(7,8)(12,21)(13,20)(14,19)(15,18)(16,17)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(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:

2,3,7,11,67|b^a=b^2,c^a=c^6,d^a=d^10,e^a=e^66

Conjugacy classes & character table

7740 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₄₇₇
maximal normal subgroup of	C₂×D₁₅₄₇₇
maximal quotient of	Dic₁₅₄₇₇, C₂×D₁₅₄₇₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30954, 25);

D15477 #44526244