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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 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4, D4 [×2], Q8 [×3], C23, D5 [×3], C10, C10, Dic3 [×2], C12, C12, D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20, C20, D10 [×4], C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, D15 [×2], D15, C30, C30, C2×SD16, C52C8, C40, Dic10, Dic10, C4×D5 [×2], 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 [×3], 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 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], 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 55 11 45)(2 54 12 44)(3 53 13 43)(4 52 14 42)(5 51 15 41)(6 50 16 60)(7 49 17 59)(8 48 18 58)(9 47 19 57)(10 46 20 56)(21 72 31 62)(22 71 32 61)(23 70 33 80)(24 69 34 79)(25 68 35 78)(26 67 36 77)(27 66 37 76)(28 65 38 75)(29 64 39 74)(30 63 40 73)(81 120 91 110)(82 119 92 109)(83 118 93 108)(84 117 94 107)(85 116 95 106)(86 115 96 105)(87 114 97 104)(88 113 98 103)(89 112 99 102)(90 111 100 101)
(1 89 36)(2 100 37 12 90 27)(3 91 38)(4 82 39 14 92 29)(5 93 40)(6 84 21 16 94 31)(7 95 22)(8 86 23 18 96 33)(9 97 24)(10 88 25 20 98 35)(11 99 26)(13 81 28)(15 83 30)(17 85 32)(19 87 34)(41 113 63 56 118 78)(42 104 64 47 119 69)(43 115 65 58 120 80)(44 106 66 49 101 71)(45 117 67 60 102 62)(46 108 68 51 103 73)(48 110 70 53 105 75)(50 112 72 55 107 77)(52 114 74 57 109 79)(54 116 76 59 111 61)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 80)(58 79)(59 78)(60 77)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)

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

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

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

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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