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G = Dic10⋊D6order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic64D10, Dic104D6, D152SD16, D30.38D4, C60.11C23, D60.3C22, Dic15.12D4, C3⋊C816D10, C53(S3×SD16), C33(D5×SD16), C52C816D6, D15⋊Q81C2, D4.D52S3, D4.S32D5, (C5×D4).3D6, C6.69(D4×D5), (C3×D4).3D10, (D4×D15).1C2, D4.13(S3×D5), C10.70(S3×D4), C1512(C2×SD16), D152C82C2, C30.173(C2×D4), Dic6⋊D52C2, C15⋊SD162C2, C20.11(C22×S3), (C5×Dic6)⋊3C22, (C4×D15).3C22, (D4×C15).5C22, C12.11(C22×D5), (C3×Dic10)⋊3C22, C2.22(D10⋊D6), C4.11(C2×S3×D5), (C5×C3⋊C8)⋊6C22, (C3×D4.D5)⋊3C2, (C5×D4.S3)⋊3C2, (C3×C52C8)⋊6C22, SmallGroup(480,563)

Series: Derived Chief Lower central Upper central

C1C60 — Dic10⋊D6
C1C5C15C30C60C3×Dic10D15⋊Q8 — Dic10⋊D6
C15C30C60 — Dic10⋊D6
C1C2C4D4

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 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B8A8B8C8D10A10B10C10D12A12B15A15B20A20B20C20D24A24B30A30B30C30D30E30F40A40B40C40D60A60B
order12222234444556688881010101012121515202020202424303030303030404040406060
size11415156022122030222866101022884404444242420204488881212121288

45 irreducible representations

dim111111112222222222244444448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6SD16D10D10D10S3×D4S3×D5D4×D5S3×SD16C2×S3×D5D5×SD16D10⋊D6Dic10⋊D6
kernelDic10⋊D6D152C8C15⋊SD16Dic6⋊D5C3×D4.D5C5×D4.S3D15⋊Q8D4×D15D4.D5Dic15D30D4.S3C52C8Dic10C5×D4D15C3⋊C8Dic6C3×D4C10D4C6C5C4C3C2C1
# reps111111111112111422212222442

Matrix representation of Dic10⋊D6 in GL6(𝔽241)

18910000
24000000
001000
000100
0000240240
000021
,
521890000
11890000
001000
000100
0000019
0000380
,
100000
010000
002395400
00174100
000011
00000240
,
189520000
240520000
00240000
00174100
000011
00000240

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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