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## G = Dic6⋊5F5order 480 = 25·3·5

### 5th semidirect product of Dic6 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic6⋊5F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — Dic3×F5 — Dic6⋊5F5
 Lower central C15 — C30 — Dic6⋊5F5
 Upper central C1 — C2 — C4

Generators and relations for Dic65F5
G = < a,b,c,d | a12=c5=d4=1, b2=a6, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 660 in 140 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×10], C22, C5, C6, C6 [×2], C2×C4 [×7], Q8 [×4], D5 [×2], C10, Dic3 [×2], Dic3 [×5], C12, C12 [×3], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5, Dic5 [×2], C20, C20 [×2], F5 [×5], D10, Dic6, Dic6 [×3], C2×Dic3 [×4], C2×C12 [×3], C3×D5 [×2], C30, C4×Q8, Dic10 [×3], C4×D5, C4×D5 [×2], C5×Q8, C2×F5 [×2], C2×F5 [×2], C4×Dic3 [×3], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C60, C3×F5 [×2], C3⋊F5 [×2], C3⋊F5, C6×D5, C4×F5 [×3], C4⋊F5, C4⋊F5 [×2], Q8×D5, Dic6⋊C4, D5×Dic3 [×2], C15⋊Q8 [×2], D5×C12, C5×Dic6, Dic30, C6×F5 [×2], C2×C3⋊F5 [×2], Q8×F5, Dic3×F5 [×2], Dic3⋊F5 [×2], C3×C4⋊F5, C4×C3⋊F5, D5×Dic6, Dic65F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D6 [×3], C22×C4, C2×Q8, C4○D4, F5, C4×S3 [×2], C22×S3, C4×Q8, C2×F5 [×3], S3×C2×C4, D42S3, S3×Q8, C22×F5, Dic6⋊C4, S3×F5, Q8×F5, C2×S3×F5, Dic65F5

Smallest permutation representation of Dic65F5
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 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 52 19 58)(14 51 20 57)(15 50 21 56)(16 49 22 55)(17 60 23 54)(18 59 24 53)(25 112 31 118)(26 111 32 117)(27 110 33 116)(28 109 34 115)(29 120 35 114)(30 119 36 113)(61 107 67 101)(62 106 68 100)(63 105 69 99)(64 104 70 98)(65 103 71 97)(66 102 72 108)(73 90 79 96)(74 89 80 95)(75 88 81 94)(76 87 82 93)(77 86 83 92)(78 85 84 91)
(1 99 15 80 109)(2 100 16 81 110)(3 101 17 82 111)(4 102 18 83 112)(5 103 19 84 113)(6 104 20 73 114)(7 105 21 74 115)(8 106 22 75 116)(9 107 23 76 117)(10 108 24 77 118)(11 97 13 78 119)(12 98 14 79 120)(25 45 66 53 86)(26 46 67 54 87)(27 47 68 55 88)(28 48 69 56 89)(29 37 70 57 90)(30 38 71 58 91)(31 39 72 59 92)(32 40 61 60 93)(33 41 62 49 94)(34 42 63 50 95)(35 43 64 51 96)(36 44 65 52 85)
(1 48 7 42)(2 43 8 37)(3 38 9 44)(4 45 10 39)(5 40 11 46)(6 47 12 41)(13 26 84 61)(14 33 73 68)(15 28 74 63)(16 35 75 70)(17 30 76 65)(18 25 77 72)(19 32 78 67)(20 27 79 62)(21 34 80 69)(22 29 81 64)(23 36 82 71)(24 31 83 66)(49 114 88 98)(50 109 89 105)(51 116 90 100)(52 111 91 107)(53 118 92 102)(54 113 93 97)(55 120 94 104)(56 115 95 99)(57 110 96 106)(58 117 85 101)(59 112 86 108)(60 119 87 103)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P 5 6A 6B 6C 10 12A 12B ··· 12F 15 20A 20B 20C 30 60A 60B order 1 2 2 2 3 4 4 4 4 ··· 4 4 4 4 4 4 4 4 4 5 6 6 6 10 12 12 ··· 12 15 20 20 20 30 60 60 size 1 1 5 5 2 2 6 6 10 ··· 10 15 15 15 15 30 30 30 30 4 2 10 10 4 4 20 ··· 20 8 8 24 24 8 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 8 8 type + + + + + + + - + + + + + - - + - + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 Q8 D6 D6 C4○D4 C4×S3 C4×S3 F5 C2×F5 C2×F5 D4⋊2S3 S3×Q8 S3×F5 Q8×F5 C2×S3×F5 Dic6⋊5F5 kernel Dic6⋊5F5 Dic3×F5 Dic3⋊F5 C3×C4⋊F5 C4×C3⋊F5 D5×Dic6 C15⋊Q8 C5×Dic6 Dic30 C4⋊F5 C3⋊F5 C4×D5 C2×F5 C3×D5 Dic5 C20 Dic6 Dic3 C12 D5 D5 C4 C3 C2 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 1 2 2 2 2 1 2 1 1 1 1 1 1 2

Matrix representation of Dic65F5 in GL8(𝔽61)

 60 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 43 38 0 0 0 0 0 0 38 18 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60
,
 1 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 0 0 0 0 60 60 60 60 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 50 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 15 9 0 0 0 0 0 0 9 46 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 0 1 1 1 1

G:=sub<GL(8,GF(61))| [60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,43,38,0,0,0,0,0,0,38,18,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,60,0,0,0],[50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,15,9,0,0,0,0,0,0,9,46,0,0,0,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,60,0,1] >;

Dic65F5 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_5F_5
% in TeX

G:=Group("Dic6:5F5");
// GroupNames label

G:=SmallGroup(480,984);
// by ID

G=gap.SmallGroup(480,984);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,219,100,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^5=d^4=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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