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G = Dic6⋊D10order 480 = 25·3·5

5th semidirect product of Dic6 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D205D6, Dic65D10, D30.39D4, C60.22C23, D60.6C22, Dic15.13D4, C3⋊C89D10, D4⋊D56S3, C52C89D6, (D4×D15)⋊2C2, C3⋊D404C2, C54(D8⋊S3), D4.S36D5, C6.72(D4×D5), C33(D40⋊C2), D4.16(S3×D5), (C5×D4).10D6, C10.73(S3×D4), D20⋊S32C2, C1520(C8⋊C22), (C3×D4).10D10, C30.184(C2×D4), Dic6⋊D54C2, (C3×D20)⋊5C22, D30.5C43C2, C20.22(C22×S3), (C5×Dic6)⋊5C22, (C4×D15).6C22, C12.22(C22×D5), (D4×C15).16C22, C2.25(D10⋊D6), C4.22(C2×S3×D5), (C3×D4⋊D5)⋊8C2, (C5×C3⋊C8)⋊9C22, (C5×D4.S3)⋊8C2, (C3×C52C8)⋊9C22, SmallGroup(480,574)

Series: Derived Chief Lower central Upper central

C1C60 — Dic6⋊D10
C1C5C15C30C60C3×D20D20⋊S3 — Dic6⋊D10
C15C30C60 — Dic6⋊D10
C1C2C4D4

Generators and relations for Dic6⋊D10
 G = < a,b,c,d | a12=c10=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >

Subgroups: 1004 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C3×D5, D15 [×2], C30, C30, C8⋊C22, C52C8, C40, C4×D5 [×2], D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, Dic15, C60, C6×D5, D30, D30 [×3], C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, D8⋊S3, C5×C3⋊C8, C3×C52C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C4×D15, D60, C157D4, D4×C15, C22×D15, D40⋊C2, D30.5C4, C3⋊D40, Dic6⋊D5, C3×D4⋊D5, C5×D4.S3, D20⋊S3, D4×D15, Dic6⋊D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D40⋊C2, D10⋊D6, Dic6⋊D10

Smallest permutation representation of Dic6⋊D10
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 37 7 43)(2 48 8 42)(3 47 9 41)(4 46 10 40)(5 45 11 39)(6 44 12 38)(13 89 19 95)(14 88 20 94)(15 87 21 93)(16 86 22 92)(17 85 23 91)(18 96 24 90)(25 103 31 97)(26 102 32 108)(27 101 33 107)(28 100 34 106)(29 99 35 105)(30 98 36 104)(49 62 55 68)(50 61 56 67)(51 72 57 66)(52 71 58 65)(53 70 59 64)(54 69 60 63)(73 115 79 109)(74 114 80 120)(75 113 81 119)(76 112 82 118)(77 111 83 117)(78 110 84 116)
(1 102 53 89 81 4 99 56 86 84)(2 97 54 96 82 11 100 51 87 79)(3 104 55 91 83 6 101 58 88 74)(5 106 57 93 73 8 103 60 90 76)(7 108 59 95 75 10 105 50 92 78)(9 98 49 85 77 12 107 52 94 80)(13 110 40 26 61)(14 117 41 33 62 20 111 47 27 68)(15 112 42 28 63)(16 119 43 35 64 22 113 37 29 70)(17 114 44 30 65)(18 109 45 25 66 24 115 39 31 72)(19 116 46 32 67)(21 118 48 34 69)(23 120 38 36 71)
(1 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 120)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 67)(62 66)(63 65)(68 72)(69 71)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 108)(93 107)(94 106)(95 105)(96 104)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C8A8B10A10B10C10D 12 15A15B20A20B20C20D24A24B30A30B30C30D30E30F40A40B40C40D60A60B
order1222223444556668810101010121515202020202424303030303030404040406060
size1142030602212302228401220228844444242420204488881212121288

42 irreducible representations

dim111111112222222222444444448
type++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C8⋊C22S3×D4S3×D5D4×D5D8⋊S3C2×S3×D5D40⋊C2D10⋊D6Dic6⋊D10
kernelDic6⋊D10D30.5C4C3⋊D40Dic6⋊D5C3×D4⋊D5C5×D4.S3D20⋊S3D4×D15D4⋊D5Dic15D30D4.S3C52C8D20C5×D4C3⋊C8Dic6C3×D4C15C10D4C6C5C4C3C2C1
# reps111111111112111222112222442

Matrix representation of Dic6⋊D10 in GL6(𝔽241)

100000
010000
000001
0000240240
00024000
001100
,
24000000
02400000
004794194147
004719419447
00194147194147
001944719447
,
1901900000
512400000
000010
000001
001000
000100
,
1901900000
240510000
000010
0000240240
001000
0024024000

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,1,0,0,0,240,0,0,0,0,1,240,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,47,194,194,0,0,94,194,147,47,0,0,194,194,194,194,0,0,147,47,147,47],[190,51,0,0,0,0,190,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[190,240,0,0,0,0,190,51,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,1,240,0,0,0,0,0,240,0,0] >;

Dic6⋊D10 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes D_{10}
% in TeX

G:=Group("Dic6:D10");
// GroupNames label

G:=SmallGroup(480,574);
// by ID

G=gap.SmallGroup(480,574);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,303,675,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^10=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

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