D₁₂₇₆₅ #38784289

order	25530
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₇₆₅
unambiguous semidirect product	C₃⋊D₄₂₅₅
unambiguous semidirect product	C₅⋊D₂₅₅₃
unambiguous semidirect product	C₂₃⋊D₅₅₅
unambiguous semidirect product	C₃₇⋊D₃₄₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₂₃×C₃₇ ⋊ C₂
Hall	C₅×C₂₃×C₃₇ ⋊ D₃
Hall	C₃×C₂₃×C₃₇ ⋊ D₅
Hall	C₂₃×C₃₇ ⋊ D₁₅
Hall	C₃×C₅×C₃₇ ⋊ D₂₃
Hall	C₃×C₅×C₂₃ ⋊ D₃₇
Hall	C₅×C₃₇ ⋊ D₆₉
Hall	C₅×C₂₃ ⋊ D₁₁₁
Hall	C₃×C₃₇ ⋊ D₁₁₅
Hall	C₃×C₂₃ ⋊ D₁₈₅
Sylow	C₃₇ ⋊ D₃₄₅
Sylow	C₂₃ ⋊ D₅₅₅
Hall	C₃×C₅ ⋊ D₈₅₁
Sylow	C₅ ⋊ D₂₅₅₃
Sylow	C₃ ⋊ D₄₂₅₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₂₃×C₃₇	C₂
LCS	G ▷ C₃×C₅×C₂₃×C₃₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 12765

A minimal faithful permutation representation — 2 generators on 68 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);(2,3)(5,8)(6,7)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2,3,5,23,37|b^a=b^2,c^a=c^4,d^a=d^22,e^a=e^36

Conjugacy classes & character table

6384 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₂₇₆₅
maximal normal subgroup of	C₂×D₁₂₇₆₅
maximal quotient of	Dic₁₂₇₆₅, C₂×D₁₂₇₆₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25530, 17);

D12765 #38784289