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G = Dic103D6order 480 = 25·3·5

3rd semidirect product of Dic10 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic103D6, C60.2C23, D12.23D10, D60.1C22, C3⋊C85D10, D4⋊S31D5, D44(S3×D5), (C5×D4)⋊2D6, D4⋊D152C2, (C3×D4)⋊8D10, (D5×D12)⋊2C2, (C6×D5).8D4, (C4×D5).5D6, D42D51S3, C37(D8⋊D5), C52(D4⋊D6), C6.139(D4×D5), C1513(C8⋊C22), C153C82C22, C20.D61C2, C15⋊SD161C2, C30.164(C2×D4), (D4×C15)⋊2C22, C20.2(C22×S3), C12.2(C22×D5), C20.32D61C2, (C3×Dic5).66D4, (D5×C12).2C22, (C5×D12).1C22, D10.17(C3⋊D4), (C3×Dic10)⋊1C22, Dic5.31(C3⋊D4), C4.2(C2×S3×D5), (C5×D4⋊S3)⋊2C2, (C5×C3⋊C8)⋊2C22, C2.21(D5×C3⋊D4), (C3×D42D5)⋊1C2, C10.42(C2×C3⋊D4), SmallGroup(480,554)

Series: Derived Chief Lower central Upper central

C1C60 — Dic103D6
C1C5C15C30C60D5×C12D5×D12 — Dic103D6
C15C30C60 — Dic103D6
C1C2C4D4

Generators and relations for Dic103D6
 G = < a,b,c,d | a20=c6=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, cbc-1=a10b, dbd=a5b, dcd=c-1 >

Subgroups: 892 in 136 conjugacy classes, 40 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 [×2], C10, C10 [×2], C12, C12 [×2], D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C3⋊C8, C3⋊C8, D12, D12 [×2], C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C4.Dic3, D4⋊S3, D4⋊S3, Q82S3 [×2], C2×D12, C3×C4○D4, C3×Dic5, C3×Dic5, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D4⋊D6, C5×C3⋊C8, C153C8, C5⋊D12, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×D12, D60, D4×C15, C2×S3×D5, D8⋊D5, C20.32D6, C20.D6, C15⋊SD16, C5×D4⋊S3, D4⋊D15, D5×D12, C3×D42D5, Dic103D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D4⋊D6, C2×S3×D5, D8⋊D5, D5×C3⋊D4, Dic103D6

Smallest permutation representation of Dic103D6
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)
(1 106 11 116)(2 105 12 115)(3 104 13 114)(4 103 14 113)(5 102 15 112)(6 101 16 111)(7 120 17 110)(8 119 18 109)(9 118 19 108)(10 117 20 107)(21 81 31 91)(22 100 32 90)(23 99 33 89)(24 98 34 88)(25 97 35 87)(26 96 36 86)(27 95 37 85)(28 94 38 84)(29 93 39 83)(30 92 40 82)(41 75 51 65)(42 74 52 64)(43 73 53 63)(44 72 54 62)(45 71 55 61)(46 70 56 80)(47 69 57 79)(48 68 58 78)(49 67 59 77)(50 66 60 76)
(1 95 51)(2 84 52 10 96 60)(3 93 53 19 97 49)(4 82 54 8 98 58)(5 91 55 17 99 47)(6 100 56)(7 89 57 15 81 45)(9 87 59 13 83 43)(11 85 41)(12 94 42 20 86 50)(14 92 44 18 88 48)(16 90 46)(21 71 110 23 69 112)(22 80 111 32 70 101)(24 78 113 30 72 119)(25 67 114 39 73 108)(26 76 115 28 74 117)(27 65 116 37 75 106)(29 63 118 35 77 104)(31 61 120 33 79 102)(34 68 103 40 62 109)(36 66 105 38 64 107)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D8A8B10A10B10C10D10E10F12A12B12C12D12E15A15B20A20B30A30B30C30D30E30F40A40B40C40D60A60B
order122222344455666688101010101010121212121215152020303030303030404040406060
size114101260221020222442012602288242441010202044444488881212121288

45 irreducible representations

dim1111111122222222222244444448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C3⋊D4C3⋊D4C8⋊C22S3×D5D4×D5D4⋊D6C2×S3×D5D8⋊D5D5×C3⋊D4Dic103D6
kernelDic103D6C20.32D6C20.D6C15⋊SD16C5×D4⋊S3D4⋊D15D5×D12C3×D42D5D42D5C3×Dic5C6×D5D4⋊S3Dic10C4×D5C5×D4C3⋊C8D12C3×D4Dic5D10C15D4C6C5C4C3C2C1
# reps1111111111121112222212222442

Matrix representation of Dic103D6 in GL6(𝔽241)

100000
010000
00521240188
002401905251
002331250240
002341240240
,
24000000
02400000
00160148225
00164014877
001516900
0020057148225
,
02400000
12400000
0051100
005119000
000010
00521189240
,
12400000
02400000
00521188240
002401905152
00002400
005112400

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,240,233,234,0,0,1,190,125,124,0,0,240,52,0,0,0,0,188,51,240,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,16,164,15,200,0,0,0,0,169,57,0,0,148,148,0,148,0,0,225,77,0,225],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,51,51,0,52,0,0,1,190,0,1,0,0,0,0,1,189,0,0,0,0,0,240],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,52,240,0,51,0,0,1,190,0,1,0,0,188,51,240,240,0,0,240,52,0,0] >;

Dic103D6 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_3D_6
% in TeX

G:=Group("Dic10:3D6");
// GroupNames label

G:=SmallGroup(480,554);
// by ID

G=gap.SmallGroup(480,554);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^6=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^10*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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