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G = C5×C8⋊D6order 480 = 25·3·5

Direct product of C5 and C8⋊D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C8⋊D6, C4019D6, D242C10, C20.66D12, C60.145D4, C12026C22, C60.271C23, C81(S3×C10), C241(C2×C10), C24⋊C21C10, C4○D122C10, (C5×D24)⋊10C2, (C2×D12)⋊7C10, D124(C2×C10), (C2×C30).92D4, C4.14(C5×D12), C12.12(C5×D4), C6.13(D4×C10), (C10×D12)⋊23C2, C1527(C8⋊C22), Dic64(C2×C10), C2.15(C10×D12), (C2×C20).240D6, C30.300(C2×D4), C10.84(C2×D12), (C2×C10).27D12, (C5×M4(2))⋊5S3, M4(2)⋊1(C5×S3), C22.5(C5×D12), (C5×D12)⋊34C22, (C3×M4(2))⋊1C10, (C15×M4(2))⋊7C2, (C2×C60).355C22, C20.235(C22×S3), C12.32(C22×C10), (C5×Dic6)⋊31C22, C31(C5×C8⋊C22), C4.32(S3×C2×C10), (C2×C6).5(C5×D4), (C5×C24⋊C2)⋊9C2, (C5×C4○D12)⋊12C2, (C2×C4).13(S3×C10), (C2×C12).28(C2×C10), SmallGroup(480,787)

Series: Derived Chief Lower central Upper central

C1C12 — C5×C8⋊D6
C1C3C6C12C60C5×D12C10×D12 — C5×C8⋊D6
C3C6C12 — C5×C8⋊D6
C1C10C2×C20C5×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 2A2B2C2D2E 3 4A4B4C5A5B5C5D6A6B8A8B10A10B10C10D10E10F10G10H10I···10T12A12B12C15A15B15C15D20A···20H20I20J20K20L24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order122222344455556688101010101010101010···101212121515151520···202020202024242424303030303030303040···4060···6060606060120···120
size11212121222212111124441111222212···1222422222···2121212124444222244444···42···244444···4

105 irreducible representations

dim111111111111222222222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D4D6D6D12D12C5×S3C5×D4C5×D4S3×C10S3×C10C5×D12C5×D12C8⋊C22C8⋊D6C5×C8⋊C22C5×C8⋊D6
kernelC5×C8⋊D6C5×C24⋊C2C5×D24C15×M4(2)C10×D12C5×C4○D12C8⋊D6C24⋊C2D24C3×M4(2)C2×D12C4○D12C5×M4(2)C60C2×C30C40C2×C20C20C2×C10M4(2)C12C2×C6C8C2×C4C4C22C15C5C3C1
# reps122111488444111212244484881248

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

98000
09800
00980
00098
,
0010
0001
994300
19814200
,
240100
240000
001240
0010
,
0100
1000
004399
00142198
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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