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## G = Dic10⋊D6order 480 = 25·3·5

### 4th semidirect product of Dic10 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic10⋊D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — D15⋊Q8 — Dic10⋊D6
 Lower central C15 — C30 — C60 — Dic10⋊D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic10⋊D6
G = < a,b,c,d | a20=c6=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a11, cbc-1=dbd=a5b, dcd=c-1 >

Subgroups: 956 in 136 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, D15, D15, C30, C30, C2×SD16, C52C8, C40, Dic10, Dic10, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, S3×SD16, C5×C3⋊C8, C3×C52C8, D30.C2, C15⋊Q8, C3×Dic10, C5×Dic6, C4×D15, D60, C157D4, D4×C15, C22×D15, D5×SD16, D152C8, C15⋊SD16, Dic6⋊D5, C3×D4.D5, C5×D4.S3, D15⋊Q8, D4×D15, Dic10⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, C2×SD16, C22×D5, S3×D4, S3×D5, D4×D5, S3×SD16, C2×S3×D5, D5×SD16, D10⋊D6, Dic10⋊D6

Smallest permutation representation of Dic10⋊D6
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 57 11 47)(2 56 12 46)(3 55 13 45)(4 54 14 44)(5 53 15 43)(6 52 16 42)(7 51 17 41)(8 50 18 60)(9 49 19 59)(10 48 20 58)(21 73 31 63)(22 72 32 62)(23 71 33 61)(24 70 34 80)(25 69 35 79)(26 68 36 78)(27 67 37 77)(28 66 38 76)(29 65 39 75)(30 64 40 74)(81 117 91 107)(82 116 92 106)(83 115 93 105)(84 114 94 104)(85 113 95 103)(86 112 96 102)(87 111 97 101)(88 110 98 120)(89 109 99 119)(90 108 100 118)
(1 62 114)(2 73 115 12 63 105)(3 64 116)(4 75 117 14 65 107)(5 66 118)(6 77 119 16 67 109)(7 68 120)(8 79 101 18 69 111)(9 70 102)(10 61 103 20 71 113)(11 72 104)(13 74 106)(15 76 108)(17 78 110)(19 80 112)(21 98 56 26 93 41)(22 89 57 37 94 52)(23 100 58 28 95 43)(24 91 59 39 96 54)(25 82 60 30 97 45)(27 84 42 32 99 47)(29 86 44 34 81 49)(31 88 46 36 83 51)(33 90 48 38 85 53)(35 92 50 40 87 55)
(1 104)(2 103)(3 102)(4 101)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 100)(57 99)(58 98)(59 97)(60 96)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 15A 15B 20A 20B 20C 20D 24A 24B 30A 30B 30C 30D 30E 30F 40A 40B 40C 40D 60A 60B order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 8 8 8 8 10 10 10 10 12 12 15 15 20 20 20 20 24 24 30 30 30 30 30 30 40 40 40 40 60 60 size 1 1 4 15 15 60 2 2 12 20 30 2 2 2 8 6 6 10 10 2 2 8 8 4 40 4 4 4 4 24 24 20 20 4 4 8 8 8 8 12 12 12 12 8 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 SD16 D10 D10 D10 S3×D4 S3×D5 D4×D5 S3×SD16 C2×S3×D5 D5×SD16 D10⋊D6 Dic10⋊D6 kernel Dic10⋊D6 D15⋊2C8 C15⋊SD16 Dic6⋊D5 C3×D4.D5 C5×D4.S3 D15⋊Q8 D4×D15 D4.D5 Dic15 D30 D4.S3 C5⋊2C8 Dic10 C5×D4 D15 C3⋊C8 Dic6 C3×D4 C10 D4 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of Dic10⋊D6 in GL6(𝔽241)

 189 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 0 0 0 0 2 1
,
 52 189 0 0 0 0 1 189 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 19 0 0 0 0 38 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 239 54 0 0 0 0 174 1 0 0 0 0 0 0 1 1 0 0 0 0 0 240
,
 189 52 0 0 0 0 240 52 0 0 0 0 0 0 240 0 0 0 0 0 174 1 0 0 0 0 0 0 1 1 0 0 0 0 0 240

G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,2,0,0,0,0,240,1],[52,1,0,0,0,0,189,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,38,0,0,0,0,19,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,239,174,0,0,0,0,54,1,0,0,0,0,0,0,1,0,0,0,0,0,1,240],[189,240,0,0,0,0,52,52,0,0,0,0,0,0,240,174,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,240] >;

Dic10⋊D6 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes D_6
% in TeX

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

G:=SmallGroup(480,563);
// by ID

G=gap.SmallGroup(480,563);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,135,675,346,185,80,1356,18822]);
// 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^11,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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