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## G = Dic5.Dic6order 480 = 25·3·5

### 3rd non-split extension by Dic5 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic5.Dic6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C3×C4⋊F5 — Dic5.Dic6
 Lower central C15 — C30 — C60 — Dic5.Dic6
 Upper central C1 — C2 — C4

Generators and relations for Dic5.Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=bc6, bab-1=a-1, cac-1=a3, ad=da, cbc-1=a5b, bd=db, dcd-1=a5bc-1 >

Subgroups: 404 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C3⋊C8, C2×Dic3, C2×C12, C3×D5, C30, C4.Q8, C52C8, C40, C4×D5, C2×F5, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×Dic5, C60, C3×F5, C3⋊F5, C6×D5, C8×D5, C4⋊F5, C4⋊F5, C12.Q8, C5×C3⋊C8, C153C8, D5×C12, C6×F5, C2×C3⋊F5, C40⋊C4, D5×C3⋊C8, C3×C4⋊F5, C60⋊C4, Dic5.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, SD16, F5, Dic6, C4×S3, C3⋊D4, C4.Q8, C2×F5, Dic3⋊C4, D4.S3, Q82S3, C4⋊F5, C12.Q8, S3×F5, C40⋊C4, Dic3⋊F5, Dic5.Dic6

Smallest permutation representation of Dic5.Dic6
On 120 points
Generators in S120
(1 83 89 95 77 7 113 41 47 119)(2 96 114 120 90 8 48 84 78 42)(3 109 37 43 115 9 79 85 91 73)(4 44 80 74 38 10 92 110 116 86)(5 75 93 87 81 11 117 45 39 111)(6 88 118 112 94 12 40 76 82 46)(13 19 106 72 25 53 59 31 66 100)(14 61 60 101 107 54 67 20 26 32)(15 102 68 33 49 55 27 62 108 21)(16 34 28 22 69 56 97 103 50 63)(17 23 98 64 29 57 51 35 70 104)(18 65 52 105 99 58 71 24 30 36)
(1 72 7 66)(2 67 8 61)(3 62 9 68)(4 69 10 63)(5 64 11 70)(6 71 12 65)(13 47 53 95)(14 96 54 48)(15 37 55 85)(16 86 56 38)(17 39 57 87)(18 88 58 40)(19 41 59 89)(20 90 60 42)(21 43 49 91)(22 92 50 44)(23 45 51 93)(24 94 52 46)(25 77 100 119)(26 120 101 78)(27 79 102 109)(28 110 103 80)(29 81 104 111)(30 112 105 82)(31 83 106 113)(32 114 107 84)(33 73 108 115)(34 116 97 74)(35 75 98 117)(36 118 99 76)
(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 63 66 10 7 69 72 4)(2 3 61 68 8 9 67 62)(5 71 70 6 11 65 64 12)(13 110 41 97 53 80 89 34)(14 102 90 115 54 27 42 73)(15 120 43 107 55 78 91 32)(16 100 92 113 56 25 44 83)(17 118 45 105 57 76 93 30)(18 98 94 111 58 35 46 81)(19 116 47 103 59 74 95 28)(20 108 96 109 60 33 48 79)(21 114 37 101 49 84 85 26)(22 106 86 119 50 31 38 77)(23 112 39 99 51 82 87 36)(24 104 88 117 52 29 40 75)

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

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

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

36 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 ··· 12F 15 20A 20B 30 40A 40B 40C 40D 60A 60B order 1 2 2 2 3 4 4 4 4 4 4 5 6 6 6 8 8 8 8 10 12 12 ··· 12 15 20 20 30 40 40 40 40 60 60 size 1 1 5 5 2 2 10 20 20 60 60 4 2 10 10 6 6 30 30 4 4 20 ··· 20 8 4 4 8 12 12 12 12 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 8 type + + + + + - + + - + + - + + - image C1 C2 C2 C2 C4 C4 S3 Q8 D4 D6 SD16 Dic6 C4×S3 C3⋊D4 F5 C2×F5 D4.S3 Q8⋊2S3 C4⋊F5 C40⋊C4 S3×F5 Dic3⋊F5 Dic5.Dic6 kernel Dic5.Dic6 D5×C3⋊C8 C3×C4⋊F5 C60⋊C4 C5×C3⋊C8 C15⋊3C8 C4⋊F5 C3×Dic5 C6×D5 C4×D5 C3×D5 Dic5 C20 D10 C3⋊C8 C12 D5 D5 C6 C3 C4 C2 C1 # reps 1 1 1 1 2 2 1 1 1 1 4 2 2 2 1 1 1 1 2 4 1 1 2

Matrix representation of Dic5.Dic6 in GL8(𝔽241)

 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 240 240
,
 156 0 5 236 0 0 0 0 0 154 3 0 0 0 0 0 0 208 87 0 0 0 0 0 192 208 2 85 0 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0
,
 78 29 143 194 0 0 0 0 172 173 163 212 0 0 0 0 69 181 122 187 0 0 0 0 143 44 63 109 0 0 0 0 0 0 0 0 0 34 17 34 0 0 0 0 224 0 207 207 0 0 0 0 207 207 0 224 0 0 0 0 34 17 34 0
,
 19 157 194 51 0 0 0 0 233 57 49 78 0 0 0 0 6 49 126 67 0 0 0 0 72 90 115 39 0 0 0 0 0 0 0 0 224 0 207 207 0 0 0 0 34 17 34 0 0 0 0 0 0 34 17 34 0 0 0 0 207 207 0 224

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[156,0,0,192,0,0,0,0,0,154,208,208,0,0,0,0,5,3,87,2,0,0,0,0,236,0,0,85,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0],[78,172,69,143,0,0,0,0,29,173,181,44,0,0,0,0,143,163,122,63,0,0,0,0,194,212,187,109,0,0,0,0,0,0,0,0,0,224,207,34,0,0,0,0,34,0,207,17,0,0,0,0,17,207,0,34,0,0,0,0,34,207,224,0],[19,233,6,72,0,0,0,0,157,57,49,90,0,0,0,0,194,49,126,115,0,0,0,0,51,78,67,39,0,0,0,0,0,0,0,0,224,34,0,207,0,0,0,0,0,17,34,207,0,0,0,0,207,34,17,0,0,0,0,0,207,0,34,224] >;

Dic5.Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_5.{\rm Dic}_6
% in TeX

G:=Group("Dic5.Dic6");
// GroupNames label

G:=SmallGroup(480,235);
// by ID

G=gap.SmallGroup(480,235);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,675,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=b*c^6,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*b*c^-1>;
// generators/relations

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