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## G = Dic10⋊Dic3order 480 = 25·3·5

### 1st semidirect product of Dic10 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic10⋊Dic3
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D5×C12 — C12.F5 — Dic10⋊Dic3
 Lower central C15 — C30 — C60 — Dic10⋊Dic3
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic10⋊Dic3
G = < a,b,c,d | a20=c6=1, b2=a10, d2=c3, bab-1=a-1, cac-1=a9, dad-1=a13, cbc-1=a10b, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 460 in 88 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C4.Dic3, C4×Dic3, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C3⋊F5, C6×D5, C2×C30, C4.F5, C4×F5, D42D5, Q83Dic3, C15⋊C8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, C2×C3⋊F5, D4⋊F5, C12.F5, C4×C3⋊F5, C3×D42D5, Dic10⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C4≀C2, C2×F5, C6.D4, C3⋊F5, C22⋊F5, Q83Dic3, C2×C3⋊F5, D4⋊F5, D10.D6, Dic10⋊Dic3

Smallest permutation representation of Dic10⋊Dic3
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 70 11 80)(2 69 12 79)(3 68 13 78)(4 67 14 77)(5 66 15 76)(6 65 16 75)(7 64 17 74)(8 63 18 73)(9 62 19 72)(10 61 20 71)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)(81 109 91 119)(82 108 92 118)(83 107 93 117)(84 106 94 116)(85 105 95 115)(86 104 96 114)(87 103 97 113)(88 102 98 112)(89 101 99 111)(90 120 100 110)
(1 90 31)(2 99 32 10 91 40)(3 88 33 19 92 29)(4 97 34 8 93 38)(5 86 35 17 94 27)(6 95 36)(7 84 37 15 96 25)(9 82 39 13 98 23)(11 100 21)(12 89 22 20 81 30)(14 87 24 18 83 28)(16 85 26)(41 68 112 47 62 118)(42 77 113 56 63 107)(43 66 114 45 64 116)(44 75 115 54 65 105)(46 73 117 52 67 103)(48 71 119 50 69 101)(49 80 120 59 70 110)(51 78 102 57 72 108)(53 76 104 55 74 106)(58 61 109 60 79 111)
(1 11)(2 8 10 4)(3 5 19 17)(6 16)(7 13 15 9)(12 18 20 14)(21 90)(22 87 30 83)(23 84 39 96)(24 81 28 89)(25 98 37 82)(26 95)(27 92 35 88)(29 86 33 94)(31 100)(32 97 40 93)(34 91 38 99)(36 85)(41 109 47 111)(42 106 56 104)(43 103 45 117)(44 120 54 110)(46 114 52 116)(48 108 50 102)(49 105 59 115)(51 119 57 101)(53 113 55 107)(58 118 60 112)(61 62 79 68)(63 76 77 74)(64 73 66 67)(65 70 75 80)(69 78 71 72)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 type + + + + + + + - + - + + + - image C1 C2 C2 C2 C4 C4 S3 D4 D4 Dic3 D6 Dic3 C3⋊D4 C3⋊D4 C4≀C2 F5 C2×F5 C3⋊F5 C22⋊F5 Q8⋊3Dic3 C2×C3⋊F5 D10.D6 D4⋊F5 Dic10⋊Dic3 kernel Dic10⋊Dic3 C12.F5 C4×C3⋊F5 C3×D4⋊2D5 C3×Dic10 D4×C15 D4⋊2D5 C3×Dic5 C6×D5 Dic10 C4×D5 C5×D4 Dic5 D10 C15 C3×D4 C12 D4 C6 C5 C4 C2 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 2 4 1 1 2 2 2 2 4 1 2

Matrix representation of Dic10⋊Dic3 in GL6(𝔽241)

 177 0 0 0 0 0 0 64 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 64 0 0 0 0 64 0 0 0 0 0 0 0 1 0 0 0 0 0 240 240 240 240 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 240 0 0 0 0 0 0 126 0 12 12 0 0 115 127 127 115 0 0 12 12 0 126 0 0 0 229 114 229
,
 240 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 240 240 240 240

G:=sub<GL(6,GF(241))| [177,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,64,0,0,0,0,64,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,240,0,1,0,0,0,240,1,0],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,126,115,12,0,0,0,0,127,12,229,0,0,12,127,0,114,0,0,12,115,126,229],[240,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;

Dic10⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,313);
// by ID

G=gap.SmallGroup(480,313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,2693,14118,4724]);
// 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^13,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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