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## G = C5×C8⋊D6order 480 = 25·3·5

### Direct product of C5 and C8⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C8⋊D6
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C10×D12 — C5×C8⋊D6
 Lower central C3 — C6 — C12 — C5×C8⋊D6
 Upper central C1 — C10 — C2×C20 — C5×M4(2)

Generators and relations for C5×C8⋊D6
G = < a,b,c,d | a5=b8=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 420 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C10, C10 [×4], Dic3, C12 [×2], D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, C2×C10, C2×C10 [×5], C24 [×2], Dic6, C4×S3, D12, D12 [×2], D12, C3⋊D4, C2×C12, C22×S3, C5×S3 [×3], C30, C30, C8⋊C22, C40 [×2], C2×C20, C2×C20, C5×D4 [×5], C5×Q8, C22×C10, C24⋊C2 [×2], D24 [×2], C3×M4(2), C2×D12, C4○D12, C5×Dic3, C60 [×2], S3×C10 [×5], C2×C30, C5×M4(2), C5×D8 [×2], C5×SD16 [×2], D4×C10, C5×C4○D4, C8⋊D6, C120 [×2], C5×Dic6, S3×C20, C5×D12, C5×D12 [×2], C5×D12, C5×C3⋊D4, C2×C60, S3×C2×C10, C5×C8⋊C22, C5×C24⋊C2 [×2], C5×D24 [×2], C15×M4(2), C10×D12, C5×C4○D12, C5×C8⋊D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C8⋊C22, C5×D4 [×2], C22×C10, C2×D12, S3×C10 [×3], D4×C10, C8⋊D6, C5×D12 [×2], S3×C2×C10, C5×C8⋊C22, C10×D12, C5×C8⋊D6

Smallest permutation representation of C5×C8⋊D6
On 120 points
Generators in S120
(1 61 26 51 93)(2 62 27 52 94)(3 63 28 53 95)(4 64 29 54 96)(5 57 30 55 89)(6 58 31 56 90)(7 59 32 49 91)(8 60 25 50 92)(9 85 110 22 36)(10 86 111 23 37)(11 87 112 24 38)(12 88 105 17 39)(13 81 106 18 40)(14 82 107 19 33)(15 83 108 20 34)(16 84 109 21 35)(41 115 74 99 65)(42 116 75 100 66)(43 117 76 101 67)(44 118 77 102 68)(45 119 78 103 69)(46 120 79 104 70)(47 113 80 97 71)(48 114 73 98 72)
(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)
(1 103 18)(2 100 19 6 104 23)(3 97 20)(4 102 21 8 98 17)(5 99 22)(7 101 24)(9 30 41)(10 27 42 14 31 46)(11 32 43)(12 29 44 16 25 48)(13 26 45)(15 28 47)(33 58 70 37 62 66)(34 63 71)(35 60 72 39 64 68)(36 57 65)(38 59 67)(40 61 69)(49 117 87)(50 114 88 54 118 84)(51 119 81)(52 116 82 56 120 86)(53 113 83)(55 115 85)(73 105 96 77 109 92)(74 110 89)(75 107 90 79 111 94)(76 112 91)(78 106 93)(80 108 95)
(1 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 32)(16 31)(33 60)(34 59)(35 58)(36 57)(37 64)(38 63)(39 62)(40 61)(42 48)(43 47)(44 46)(49 83)(50 82)(51 81)(52 88)(53 87)(54 86)(55 85)(56 84)(66 72)(67 71)(68 70)(73 75)(76 80)(77 79)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 112)(96 111)(97 101)(98 100)(102 104)(113 117)(114 116)(118 120)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10T 12A 12B 12C 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 2 2 3 4 4 4 5 5 5 5 6 6 8 8 10 10 10 10 10 10 10 10 10 ··· 10 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 12 12 12 2 2 2 12 1 1 1 1 2 4 4 4 1 1 1 1 2 2 2 2 12 ··· 12 2 2 4 2 2 2 2 2 ··· 2 12 12 12 12 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 D12 D12 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×D12 C5×D12 C8⋊C22 C8⋊D6 C5×C8⋊C22 C5×C8⋊D6 kernel C5×C8⋊D6 C5×C24⋊C2 C5×D24 C15×M4(2) C10×D12 C5×C4○D12 C8⋊D6 C24⋊C2 D24 C3×M4(2) C2×D12 C4○D12 C5×M4(2) C60 C2×C30 C40 C2×C20 C20 C2×C10 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C15 C5 C3 C1 # reps 1 2 2 1 1 1 4 8 8 4 4 4 1 1 1 2 1 2 2 4 4 4 8 4 8 8 1 2 4 8

Matrix representation of C5×C8⋊D6 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 98 0 0 0 0 98
,
 0 0 1 0 0 0 0 1 99 43 0 0 198 142 0 0
,
 240 1 0 0 240 0 0 0 0 0 1 240 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 43 99 0 0 142 198
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,0,99,198,0,0,43,142,1,0,0,0,0,1,0,0],[240,240,0,0,1,0,0,0,0,0,1,1,0,0,240,0],[0,1,0,0,1,0,0,0,0,0,43,142,0,0,99,198] >;

C5×C8⋊D6 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes D_6
% in TeX

G:=Group("C5xC8:D6");
// GroupNames label

G:=SmallGroup(480,787);
// by ID

G=gap.SmallGroup(480,787);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,891,226,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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