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## G = Dic6⋊D10order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic6⋊D10
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — D20⋊S3 — Dic6⋊D10
 Lower central C15 — C30 — C60 — Dic6⋊D10
 Upper central C1 — C2 — C4 — D4

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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C22×S3, C3×D5, D15, C30, C30, C8⋊C22, C52C8, C40, C4×D5, D20, D20, 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, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, 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 85 7 91)(2 96 8 90)(3 95 9 89)(4 94 10 88)(5 93 11 87)(6 92 12 86)(13 97 19 103)(14 108 20 102)(15 107 21 101)(16 106 22 100)(17 105 23 99)(18 104 24 98)(25 43 31 37)(26 42 32 48)(27 41 33 47)(28 40 34 46)(29 39 35 45)(30 38 36 44)(49 74 55 80)(50 73 56 79)(51 84 57 78)(52 83 58 77)(53 82 59 76)(54 81 60 75)(61 113 67 119)(62 112 68 118)(63 111 69 117)(64 110 70 116)(65 109 71 115)(66 120 72 114)
(1 83 63 42 14 4 80 66 39 17)(2 78 64 37 15 11 81 61 40 24)(3 73 65 44 16 6 82 68 41 19)(5 75 67 46 18 8 84 70 43 21)(7 77 69 48 20 10 74 72 45 23)(9 79 71 38 22 12 76 62 47 13)(25 104 87 57 113 31 98 93 51 119)(26 99 88 52 114)(27 106 89 59 115 33 100 95 53 109)(28 101 90 54 116)(29 108 91 49 117 35 102 85 55 111)(30 103 92 56 118)(32 105 94 58 120)(34 107 96 60 110)(36 97 86 50 112)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 60)(37 73)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(109 113)(110 112)(114 120)(115 119)(116 118)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 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 D10 D10 D10 C8⋊C22 S3×D4 S3×D5 D4×D5 D8⋊S3 C2×S3×D5 D40⋊C2 D10⋊D6 Dic6⋊D10 kernel Dic6⋊D10 D30.5C4 C3⋊D40 Dic6⋊D5 C3×D4⋊D5 C5×D4.S3 D20⋊S3 D4×D15 D4⋊D5 Dic15 D30 D4.S3 C5⋊2C8 D20 C5×D4 C3⋊C8 Dic6 C3×D4 C15 C10 D4 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 1 1 2 2 2 2 4 4 2

Matrix representation of Dic6⋊D10 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 0 0 0 240 0 0 0 0 1 1 0 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 47 94 194 147 0 0 47 194 194 47 0 0 194 147 194 147 0 0 194 47 194 47
,
 190 190 0 0 0 0 51 240 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
,
 190 190 0 0 0 0 240 51 0 0 0 0 0 0 0 0 1 0 0 0 0 0 240 240 0 0 1 0 0 0 0 0 240 240 0 0

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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