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## G = Dic5.4Dic6order 480 = 25·3·5

### 4th non-split extension by Dic5 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic5.4Dic6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C3×C4⋊F5 — Dic5.4Dic6
 Lower central C15 — C30 — C60 — Dic5.4Dic6
 Upper central C1 — C2 — C4

Generators and relations for Dic5.4Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=a5bc6, 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, C2.D8, 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, C6.Q16, C5×C3⋊C8, C153C8, D5×C12, C6×F5, C2×C3⋊F5, D5.D8, D5×C3⋊C8, C3×C4⋊F5, C60⋊C4, Dic5.4Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D8, Q16, F5, Dic6, C4×S3, C3⋊D4, C2.D8, C2×F5, Dic3⋊C4, D4⋊S3, C3⋊Q16, C4⋊F5, C6.Q16, S3×F5, D5.D8, Dic3⋊F5, Dic5.4Dic6

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

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

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

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

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 2 4 4 4 4 4 4 8 8 8 type + + + + + - + + + - - + + + - + - image C1 C2 C2 C2 C4 C4 S3 Q8 D4 D6 D8 Q16 Dic6 C4×S3 C3⋊D4 F5 C2×F5 D4⋊S3 C3⋊Q16 C4⋊F5 D5.D8 S3×F5 Dic3⋊F5 Dic5.4Dic6 kernel Dic5.4Dic6 D5×C3⋊C8 C3×C4⋊F5 C60⋊C4 C5×C3⋊C8 C15⋊3C8 C4⋊F5 C3×Dic5 C6×D5 C4×D5 C3×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 2 2 2 2 2 1 1 1 1 2 4 1 1 2

Matrix representation of Dic5.4Dic6 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
,
 0 0 1 0 0 0 0 0 0 0 0 1 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 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
,
 10 12 231 62 0 0 0 0 201 210 195 220 0 0 0 0 231 62 231 229 0 0 0 0 195 220 40 31 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
,
 220 179 210 229 0 0 0 0 86 21 86 31 0 0 0 0 31 12 220 179 0 0 0 0 155 210 86 21 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],[0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0],[10,201,231,195,0,0,0,0,12,210,62,220,0,0,0,0,231,195,231,40,0,0,0,0,62,220,229,31,0,0,0,0,0,0,0,0,0,224,207,34,0,0,0,0,34,0,207,17,0,0,0,0,17,207,0,34,0,0,0,0,34,207,224,0],[220,86,31,155,0,0,0,0,179,21,12,210,0,0,0,0,210,86,220,86,0,0,0,0,229,31,179,21,0,0,0,0,0,0,0,0,224,34,0,207,0,0,0,0,0,17,34,207,0,0,0,0,207,34,17,0,0,0,0,0,207,0,34,224] >;

Dic5.4Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_5._4{\rm Dic}_6
% in TeX

G:=Group("Dic5.4Dic6");
// GroupNames label

G:=SmallGroup(480,236);
// by ID

G=gap.SmallGroup(480,236);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=a^5*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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