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## G = Dic5.D12order 480 = 25·3·5

### 3rd non-split extension by Dic5 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic5.D12
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C6×Dic5 — C3×C22.F5 — Dic5.D12
 Lower central C15 — C30 — C2×C30 — Dic5.D12
 Upper central C1 — C2 — C22

Generators and relations for Dic5.D12
G = < a,b,c,d | a10=1, b2=c12=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, cbc-1=a5b, bd=db, dcd-1=a5bc11 >

Subgroups: 660 in 92 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C15, M4(2), C2×D4, Dic5, D10, C2×C10, C2×C10, C3⋊C8, C24, D12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C30, C4.D4, C5⋊C8, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C4.Dic3, C3×M4(2), C2×D12, C3×Dic5, S3×C10, D30, C2×C30, C22.F5, C22.F5, C2×C5⋊D4, C12.46D4, C3×C5⋊C8, C15⋊C8, C5⋊D12, C6×Dic5, S3×C2×C10, C22×D15, C23.F5, C3×C22.F5, C158M4(2), C2×C5⋊D12, Dic5.D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C4.D4, C2×F5, D6⋊C4, C22⋊F5, C12.46D4, S3×F5, C23.F5, D6⋊F5, Dic5.D12

Smallest permutation representation of Dic5.D12
On 120 points
Generators in S120
(1 98 58 32 75 13 110 70 44 87)(2 33 111 88 59 14 45 99 76 71)(3 89 46 72 112 15 77 34 60 100)(4 49 78 101 47 16 61 90 113 35)(5 102 62 36 79 17 114 50 48 91)(6 37 115 92 63 18 25 103 80 51)(7 93 26 52 116 19 81 38 64 104)(8 53 82 105 27 20 65 94 117 39)(9 106 66 40 83 21 118 54 28 95)(10 41 119 96 67 22 29 107 84 55)(11 73 30 56 120 23 85 42 68 108)(12 57 86 109 31 24 69 74 97 43)
(1 19 13 7)(2 8 14 20)(3 21 15 9)(4 10 16 22)(5 23 17 11)(6 12 18 24)(25 31 37 43)(26 44 38 32)(27 33 39 45)(28 46 40 34)(29 35 41 47)(30 48 42 36)(49 55 61 67)(50 68 62 56)(51 57 63 69)(52 70 64 58)(53 59 65 71)(54 72 66 60)(73 91 85 79)(74 80 86 92)(75 93 87 81)(76 82 88 94)(77 95 89 83)(78 84 90 96)(97 103 109 115)(98 116 110 104)(99 105 111 117)(100 118 112 106)(101 107 113 119)(102 120 114 108)
(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 6 19 12 13 18 7 24)(2 23 8 17 14 11 20 5)(3 4 21 10 15 16 9 22)(25 38 31 32 37 26 43 44)(27 36 33 30 39 48 45 42)(28 41 46 47 40 29 34 35)(49 66 55 60 61 54 67 72)(50 71 68 53 62 59 56 65)(51 64 57 58 63 52 69 70)(73 82 91 88 85 94 79 76)(74 75 80 93 86 87 92 81)(77 78 95 84 89 90 83 96)(97 98 103 116 109 110 115 104)(99 120 105 114 111 108 117 102)(100 101 118 107 112 113 106 119)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 5 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 12A 12B 12C 15 24A 24B 24C 24D 30A 30B 30C order 1 2 2 2 2 3 4 4 5 6 6 8 8 8 8 10 10 10 10 10 10 10 12 12 12 15 24 24 24 24 30 30 30 size 1 1 2 12 60 2 10 10 4 2 4 20 20 60 60 4 4 4 12 12 12 12 10 10 20 8 20 20 20 20 8 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 8 8 8 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D6 D12 C3⋊D4 C4×S3 F5 C4.D4 C2×F5 C22⋊F5 C12.46D4 C23.F5 S3×F5 D6⋊F5 Dic5.D12 kernel Dic5.D12 C3×C22.F5 C15⋊8M4(2) C2×C5⋊D12 S3×C2×C10 C22×D15 C22.F5 C3×Dic5 C2×Dic5 Dic5 Dic5 C2×C10 C22×S3 C15 C2×C6 C6 C5 C3 C22 C2 C1 # reps 1 1 1 1 2 2 1 2 1 2 2 2 1 1 1 2 2 4 1 1 2

Matrix representation of Dic5.D12 in GL8(𝔽241)

 91 0 0 0 0 0 0 0 129 240 1 0 0 0 0 0 239 188 52 0 0 0 0 0 36 227 120 98 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 0 240
,
 125 39 141 120 0 0 0 0 109 74 42 186 0 0 0 0 205 230 205 186 0 0 0 0 196 129 177 78 0 0 0 0 0 0 0 0 56 74 0 0 0 0 0 0 26 185 0 0 0 0 0 0 189 155 142 43 0 0 0 0 0 155 198 99
,
 81 128 224 0 0 0 0 0 135 223 32 224 0 0 0 0 105 146 180 191 0 0 0 0 167 101 210 239 0 0 0 0 0 0 0 0 4 0 97 97 0 0 0 0 10 0 2 1 0 0 0 0 67 142 239 239 0 0 0 0 54 198 239 239
,
 81 128 224 0 0 0 0 0 135 223 32 224 0 0 0 0 105 146 180 191 0 0 0 0 167 101 210 239 0 0 0 0 0 0 0 0 4 0 97 97 0 0 0 0 10 0 1 2 0 0 0 0 67 142 239 239 0 0 0 0 80 142 239 239

G:=sub<GL(8,GF(241))| [91,129,239,36,0,0,0,0,0,240,188,227,0,0,0,0,0,1,52,120,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[125,109,205,196,0,0,0,0,39,74,230,129,0,0,0,0,141,42,205,177,0,0,0,0,120,186,186,78,0,0,0,0,0,0,0,0,56,26,189,0,0,0,0,0,74,185,155,155,0,0,0,0,0,0,142,198,0,0,0,0,0,0,43,99],[81,135,105,167,0,0,0,0,128,223,146,101,0,0,0,0,224,32,180,210,0,0,0,0,0,224,191,239,0,0,0,0,0,0,0,0,4,10,67,54,0,0,0,0,0,0,142,198,0,0,0,0,97,2,239,239,0,0,0,0,97,1,239,239],[81,135,105,167,0,0,0,0,128,223,146,101,0,0,0,0,224,32,180,210,0,0,0,0,0,224,191,239,0,0,0,0,0,0,0,0,4,10,67,80,0,0,0,0,0,0,142,142,0,0,0,0,97,1,239,239,0,0,0,0,97,2,239,239] >;

Dic5.D12 in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_{12}
% in TeX

G:=Group("Dic5.D12");
// GroupNames label

G:=SmallGroup(480,250);
// by ID

G=gap.SmallGroup(480,250);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^10=1,b^2=c^12=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*b*c^11>;
// generators/relations

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