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## G = Dic10⋊2Dic3order 480 = 25·3·5

### 2nd semidirect product of Dic10 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic10⋊2Dic3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C60⋊C4 — Dic10⋊2Dic3
 Lower central C15 — C30 — C60 — Dic10⋊2Dic3
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Dic102Dic3
G = < a,b,c,d | a20=c6=1, b2=a10, d2=c3, bab-1=a-1, cac-1=a9, dad-1=a7, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 444 in 84 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, Q8, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C3⋊C8, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C3×D5, C30, Q8⋊C4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, C3⋊F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, Q82Dic3, C15⋊C8, C3×Dic10, C3×Dic10, D5×C12, D5×C12, Q8×C15, C2×C3⋊F5, Q8⋊F5, C60.C4, C60⋊C4, C3×Q8×D5, Dic102Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, F5, C2×Dic3, C3⋊D4, Q8⋊C4, C2×F5, Q82S3, C3⋊Q16, C6.D4, C3⋊F5, C22⋊F5, Q82Dic3, C2×C3⋊F5, Q8⋊F5, D10.D6, Dic102Dic3

Smallest permutation representation of Dic102Dic3
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 44 11 54)(2 43 12 53)(3 42 13 52)(4 41 14 51)(5 60 15 50)(6 59 16 49)(7 58 17 48)(8 57 18 47)(9 56 19 46)(10 55 20 45)(21 84 31 94)(22 83 32 93)(23 82 33 92)(24 81 34 91)(25 100 35 90)(26 99 36 89)(27 98 37 88)(28 97 38 87)(29 96 39 86)(30 95 40 85)(61 104 71 114)(62 103 72 113)(63 102 73 112)(64 101 74 111)(65 120 75 110)(66 119 76 109)(67 118 77 108)(68 117 78 107)(69 116 79 106)(70 115 80 105)
(1 103 96)(2 112 97 10 104 85)(3 101 98 19 105 94)(4 110 99 8 106 83)(5 119 100 17 107 92)(6 108 81)(7 117 82 15 109 90)(9 115 84 13 111 88)(11 113 86)(12 102 87 20 114 95)(14 120 89 18 116 93)(16 118 91)(21 42 74 37 46 70)(22 51 75 26 47 79)(23 60 76 35 48 68)(24 49 77)(25 58 78 33 50 66)(27 56 80 31 52 64)(28 45 61 40 53 73)(29 54 62)(30 43 63 38 55 71)(32 41 65 36 57 69)(34 59 67)(39 44 72)
(2 4 10 8)(3 7 19 15)(5 13 17 9)(6 16)(12 14 20 18)(21 63 37 71)(22 66 26 78)(23 69 35 65)(24 72)(25 75 33 79)(27 61 31 73)(28 64 40 80)(29 67)(30 70 38 74)(32 76 36 68)(34 62)(39 77)(41 60 57 48)(42 43 46 55)(44 49)(45 52 53 56)(47 58 51 50)(54 59)(81 118)(82 101 90 105)(83 104 99 112)(84 107 88 119)(85 110 97 106)(86 113)(87 116 95 120)(89 102 93 114)(91 108)(92 111 100 115)(94 117 98 109)(96 103)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 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 8 type + + + + + + + - + - - + + + - + - image C1 C2 C2 C2 C4 C4 S3 D4 D4 Dic3 D6 Dic3 SD16 Q16 C3⋊D4 C3⋊D4 F5 C2×F5 Q8⋊2S3 C3⋊Q16 C3⋊F5 C22⋊F5 C2×C3⋊F5 D10.D6 Q8⋊F5 Dic10⋊2Dic3 kernel Dic10⋊2Dic3 C60.C4 C60⋊C4 C3×Q8×D5 C3×Dic10 Q8×C15 Q8×D5 C3×Dic5 C6×D5 Dic10 C4×D5 C5×Q8 C3×D5 C3×D5 Dic5 D10 C3×Q8 C12 D5 D5 Q8 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 4 1 2

Matrix representation of Dic102Dic3 in GL6(𝔽241)

 240 3 0 0 0 0 160 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 0 184 0 0 0 0 148 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240 0 0 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 12 12 0 126 0 0 0 229 114 229 0 0 229 114 229 0 0 0 126 0 12 12
,
 177 0 0 0 0 0 118 64 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0

G:=sub<GL(6,GF(241))| [240,160,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,148,0,0,0,0,184,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,12,0,229,126,0,0,12,229,114,0,0,0,0,114,229,12,0,0,126,229,0,12],[177,118,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0] >;

Dic102Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("Dic10:2Dic3");
// GroupNames label

G:=SmallGroup(480,314);
// by ID

G=gap.SmallGroup(480,314);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,675,346,80,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^6=1,b^2=a^10,d^2=c^3,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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