Copied to
clipboard

G = D10⋊Dic3order 240 = 24·3·5

1st semidirect product of D10 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.10D20, C30.13D4, D101Dic3, (C6×D5)⋊1C4, C6.13(C4×D5), (C2×C6).6D10, (C2×C10).6D6, C154(C22⋊C4), C30.29(C2×C4), (C2×Dic3)⋊1D5, C2.4(D5×Dic3), C33(D10⋊C4), C22.5(S3×D5), (C10×Dic3)⋊1C2, (C2×Dic15)⋊5C2, C2.1(C15⋊D4), C52(C6.D4), C2.1(C3⋊D20), C6.11(C5⋊D4), (C2×C30).3C22, (C22×D5).2S3, C10.11(C3⋊D4), C10.11(C2×Dic3), (D5×C2×C6).1C2, SmallGroup(240,26)

Series: Derived Chief Lower central Upper central

C1C30 — D10⋊Dic3
C1C5C15C30C2×C30D5×C2×C6 — D10⋊Dic3
C15C30 — D10⋊Dic3
C1C22

Generators and relations for D10⋊Dic3
 G = < a,b,c,d | a10=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 272 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4 [×2], C23, D5 [×2], C10 [×3], Dic3 [×2], C2×C6, C2×C6 [×4], C15, C22⋊C4, Dic5, C20, D10 [×2], D10 [×2], C2×C10, C2×Dic3, C2×Dic3, C22×C6, C3×D5 [×2], C30 [×3], C2×Dic5, C2×C20, C22×D5, C6.D4, C5×Dic3, Dic15, C6×D5 [×2], C6×D5 [×2], C2×C30, D10⋊C4, C10×Dic3, C2×Dic15, D5×C2×C6, D10⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, D5×Dic3, C15⋊D4, C3⋊D20, D10⋊Dic3

Smallest permutation representation of D10⋊Dic3
On 120 points
Generators in S120
(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)(111 112 113 114 115 116 117 118 119 120)
(1 52)(2 51)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 99)(12 98)(13 97)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 100)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 50)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 70)(40 69)(71 103)(72 102)(73 101)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 120)
(1 65 21 53 35 50)(2 66 22 54 36 41)(3 67 23 55 37 42)(4 68 24 56 38 43)(5 69 25 57 39 44)(6 70 26 58 40 45)(7 61 27 59 31 46)(8 62 28 60 32 47)(9 63 29 51 33 48)(10 64 30 52 34 49)(11 83 110 95 113 80)(12 84 101 96 114 71)(13 85 102 97 115 72)(14 86 103 98 116 73)(15 87 104 99 117 74)(16 88 105 100 118 75)(17 89 106 91 119 76)(18 90 107 92 120 77)(19 81 108 93 111 78)(20 82 109 94 112 79)
(1 113 53 83)(2 114 54 84)(3 115 55 85)(4 116 56 86)(5 117 57 87)(6 118 58 88)(7 119 59 89)(8 120 60 90)(9 111 51 81)(10 112 52 82)(11 65 95 35)(12 66 96 36)(13 67 97 37)(14 68 98 38)(15 69 99 39)(16 70 100 40)(17 61 91 31)(18 62 92 32)(19 63 93 33)(20 64 94 34)(21 110 50 80)(22 101 41 71)(23 102 42 72)(24 103 43 73)(25 104 44 74)(26 105 45 75)(27 106 46 76)(28 107 47 77)(29 108 48 78)(30 109 49 79)

G:=sub<Sym(120)| (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)(111,112,113,114,115,116,117,118,119,120), (1,52)(2,51)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,100)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,50)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,70)(40,69)(71,103)(72,102)(73,101)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,120), (1,65,21,53,35,50)(2,66,22,54,36,41)(3,67,23,55,37,42)(4,68,24,56,38,43)(5,69,25,57,39,44)(6,70,26,58,40,45)(7,61,27,59,31,46)(8,62,28,60,32,47)(9,63,29,51,33,48)(10,64,30,52,34,49)(11,83,110,95,113,80)(12,84,101,96,114,71)(13,85,102,97,115,72)(14,86,103,98,116,73)(15,87,104,99,117,74)(16,88,105,100,118,75)(17,89,106,91,119,76)(18,90,107,92,120,77)(19,81,108,93,111,78)(20,82,109,94,112,79), (1,113,53,83)(2,114,54,84)(3,115,55,85)(4,116,56,86)(5,117,57,87)(6,118,58,88)(7,119,59,89)(8,120,60,90)(9,111,51,81)(10,112,52,82)(11,65,95,35)(12,66,96,36)(13,67,97,37)(14,68,98,38)(15,69,99,39)(16,70,100,40)(17,61,91,31)(18,62,92,32)(19,63,93,33)(20,64,94,34)(21,110,50,80)(22,101,41,71)(23,102,42,72)(24,103,43,73)(25,104,44,74)(26,105,45,75)(27,106,46,76)(28,107,47,77)(29,108,48,78)(30,109,49,79)>;

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)(111,112,113,114,115,116,117,118,119,120), (1,52)(2,51)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,100)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,50)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,70)(40,69)(71,103)(72,102)(73,101)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,120), (1,65,21,53,35,50)(2,66,22,54,36,41)(3,67,23,55,37,42)(4,68,24,56,38,43)(5,69,25,57,39,44)(6,70,26,58,40,45)(7,61,27,59,31,46)(8,62,28,60,32,47)(9,63,29,51,33,48)(10,64,30,52,34,49)(11,83,110,95,113,80)(12,84,101,96,114,71)(13,85,102,97,115,72)(14,86,103,98,116,73)(15,87,104,99,117,74)(16,88,105,100,118,75)(17,89,106,91,119,76)(18,90,107,92,120,77)(19,81,108,93,111,78)(20,82,109,94,112,79), (1,113,53,83)(2,114,54,84)(3,115,55,85)(4,116,56,86)(5,117,57,87)(6,118,58,88)(7,119,59,89)(8,120,60,90)(9,111,51,81)(10,112,52,82)(11,65,95,35)(12,66,96,36)(13,67,97,37)(14,68,98,38)(15,69,99,39)(16,70,100,40)(17,61,91,31)(18,62,92,32)(19,63,93,33)(20,64,94,34)(21,110,50,80)(22,101,41,71)(23,102,42,72)(24,103,43,73)(25,104,44,74)(26,105,45,75)(27,106,46,76)(28,107,47,77)(29,108,48,78)(30,109,49,79) );

G=PermutationGroup([(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),(111,112,113,114,115,116,117,118,119,120)], [(1,52),(2,51),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,99),(12,98),(13,97),(14,96),(15,95),(16,94),(17,93),(18,92),(19,91),(20,100),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,50),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,62),(38,61),(39,70),(40,69),(71,103),(72,102),(73,101),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(89,111),(90,120)], [(1,65,21,53,35,50),(2,66,22,54,36,41),(3,67,23,55,37,42),(4,68,24,56,38,43),(5,69,25,57,39,44),(6,70,26,58,40,45),(7,61,27,59,31,46),(8,62,28,60,32,47),(9,63,29,51,33,48),(10,64,30,52,34,49),(11,83,110,95,113,80),(12,84,101,96,114,71),(13,85,102,97,115,72),(14,86,103,98,116,73),(15,87,104,99,117,74),(16,88,105,100,118,75),(17,89,106,91,119,76),(18,90,107,92,120,77),(19,81,108,93,111,78),(20,82,109,94,112,79)], [(1,113,53,83),(2,114,54,84),(3,115,55,85),(4,116,56,86),(5,117,57,87),(6,118,58,88),(7,119,59,89),(8,120,60,90),(9,111,51,81),(10,112,52,82),(11,65,95,35),(12,66,96,36),(13,67,97,37),(14,68,98,38),(15,69,99,39),(16,70,100,40),(17,61,91,31),(18,62,92,32),(19,63,93,33),(20,64,94,34),(21,110,50,80),(22,101,41,71),(23,102,42,72),(24,103,43,73),(25,104,44,74),(26,105,45,75),(27,106,46,76),(28,107,47,77),(29,108,48,78),(30,109,49,79)])

D10⋊Dic3 is a maximal subgroup of
Dic3⋊C4⋊D5  D10⋊Dic6  Dic3.D20  D30.34D4  D30.D4  (D5×C12)⋊C4  (C4×D5)⋊Dic3  C60.67D4  C60.68D4  (C2×C12).D10  (C2×C60).C22  (C4×Dic3)⋊D5  C60.44D4  (C4×Dic15)⋊C2  C60.88D4  (D5×Dic3)⋊C4  D10.19(C4×S3)  (C6×D5).D4  Dic15⋊D4  Dic3⋊D20  D101Dic6  D102Dic6  Dic3×D20  Dic15.D4  D104Dic6  D208Dic3  C4×C15⋊D4  D6⋊(C4×D5)  C4×C3⋊D20  C1520(C4×D4)  D6⋊C4⋊D5  D10⋊C4⋊S3  (C2×Dic6)⋊D5  C604D4  D6.9D20  Dic15.10D4  Dic15.31D4  C122D20  S3×D10⋊C4  D30.27D4  D64D20  D306D4  C6.(D4×D5)  (C2×C30).D4  C6.(C2×D20)  D5×C6.D4  C23.17(S3×D5)  Dic3×C5⋊D4  Dic1516D4  (C2×C30)⋊D4  (C2×C6)⋊8D20  (S3×C10)⋊D4  (C2×C6)⋊D20  Dic1518D4  D308D4
D10⋊Dic3 is a maximal quotient of
C60.93D4  C60.28D4  C12.6D20  C30.D8  C6.D40  C30.Q16  C6.Dic20  C60.96D4  C60.97D4  C30.24C42  (C2×C6).D20

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C6D6E6F6G10A···10F15A15B20A···20H30A···30F
order1222223444455666666610···10151520···2030···30
size11111010266303022222101010102···2446···64···4

42 irreducible representations

dim1111122222222224444
type+++++++-++++--+
imageC1C2C2C2C4S3D4D5Dic3D6D10C3⋊D4C4×D5D20C5⋊D4S3×D5D5×Dic3C15⋊D4C3⋊D20
kernelD10⋊Dic3C10×Dic3C2×Dic15D5×C2×C6C6×D5C22×D5C30C2×Dic3D10C2×C10C2×C6C10C6C6C6C22C2C2C2
# reps1111412221244442222

Matrix representation of D10⋊Dic3 in GL4(𝔽61) generated by

171700
44100
00600
00060
,
444400
601700
00600
00231
,
60000
06000
00130
002547
,
295400
73200
00275
001334
G:=sub<GL(4,GF(61))| [17,44,0,0,17,1,0,0,0,0,60,0,0,0,0,60],[44,60,0,0,44,17,0,0,0,0,60,23,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,13,25,0,0,0,47],[29,7,0,0,54,32,0,0,0,0,27,13,0,0,5,34] >;

D10⋊Dic3 in GAP, Magma, Sage, TeX

D_{10}\rtimes {\rm Dic}_3
% in TeX

G:=Group("D10:Dic3");
// GroupNames label

G:=SmallGroup(240,26);
// by ID

G=gap.SmallGroup(240,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽