C₂×D₁₅₂₁₉ #72295754

order	60876
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₀₄₃₈
direct product	C₂×D₁₅₂₁₉

Splittings

Fitting	C₂×C₉×C₁₉×C₈₉ ⋊ C₂
derived, Hall	C₉×C₁₉×C₈₉ ⋊ C₂²
Hall	C₁₉×C₈₉ ⋊ C₂×D₉
Hall	C₉×C₈₉ ⋊ C₂×D₁₉
Hall	C₉×C₁₉ ⋊ C₂×D₈₉
Sylow	C₈₉ ⋊ C₂×D₁₇₁
Sylow	C₁₉ ⋊ C₂×D₈₀₁
Sylow	C₉ ⋊ C₂×D₁₆₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₅₂₁₉
commutator, cosocle, hypocenter	C₉×C₁₉×C₈₉	C₂²
Frattini	C₃	C₂×D₅₀₇₃
Fitting	C₂×C₉×C₁₉×C₈₉	C₂
socle	C₂×C₃×C₁₉×C₈₉	D₃
LCS	G ▷ C₉×C₁₉×C₈₉
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 30438

A minimal faithful permutation representation — 2 generators on 119 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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(4,11)(5,10)(6,9)(7,8)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Conjugacy classes & character table

15222 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119), (4,11)(5,10)(6,9)(7,8)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76));

C2×D15219 #72295754