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G = C3×C8⋊D10order 480 = 25·3·5

Direct product of C3 and C8⋊D10

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

Aliases: C3×C8⋊D10, D402C6, C2416D10, C12.66D20, C60.122D4, C12016C22, C60.264C23, C81(C6×D5), C401(C2×C6), C4○D202C6, (C2×D20)⋊7C6, D204(C2×C6), C40⋊C21C6, (C3×D40)⋊10C2, (C6×D20)⋊23C2, C10.11(C6×D4), C20.12(C3×D4), C6.84(C2×D20), C2.15(C6×D20), (C2×C30).82D4, C4.14(C3×D20), (C2×C6).27D20, C1526(C8⋊C22), Dic104(C2×C6), C30.285(C2×D4), (C5×M4(2))⋊1C6, (C3×M4(2))⋊3D5, M4(2)⋊1(C3×D5), C22.5(C3×D20), (C2×C12).237D10, (C3×D20)⋊34C22, (C15×M4(2))⋊3C2, C20.31(C22×C6), (C2×C60).287C22, C12.237(C22×D5), (C3×Dic10)⋊31C22, C51(C3×C8⋊C22), C4.30(D5×C2×C6), (C3×C40⋊C2)⋊5C2, (C2×C4).11(C6×D5), (C2×C10).5(C3×D4), (C3×C4○D20)⋊12C2, (C2×C20).24(C2×C6), SmallGroup(480,701)

Series: Derived Chief Lower central Upper central

C1C20 — C3×C8⋊D10
C1C5C10C20C60C3×D20C6×D20 — C3×C8⋊D10
C5C10C20 — C3×C8⋊D10
C1C6C2×C12C3×M4(2)

Generators and relations for C3×C8⋊D10
 G = < a,b,c,d | a3=b8=c10=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 608 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, C6, C6 [×4], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×3], C10, C10, C12 [×2], C12, C2×C6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10 [×5], C2×C10, C24 [×2], C2×C12, C2×C12, C3×D4 [×5], C3×Q8, C22×C6, C3×D5 [×3], C30, C30, C8⋊C22, C40 [×2], Dic10, C4×D5, D20, D20 [×2], D20, C5⋊D4, C2×C20, C22×D5, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C6×D4, C3×C4○D4, C3×Dic5, C60 [×2], C6×D5 [×5], C2×C30, C40⋊C2 [×2], D40 [×2], C5×M4(2), C2×D20, C4○D20, C3×C8⋊C22, C120 [×2], C3×Dic10, D5×C12, C3×D20, C3×D20 [×2], C3×D20, C3×C5⋊D4, C2×C60, D5×C2×C6, C8⋊D10, C3×C40⋊C2 [×2], C3×D40 [×2], C15×M4(2), C6×D20, C3×C4○D20, C3×C8⋊D10
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C8⋊C22, D20 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×D20, C3×C8⋊C22, C3×D20 [×2], D5×C2×C6, C8⋊D10, C6×D20, C3×C8⋊D10

Smallest permutation representation of C3×C8⋊D10
On 120 points
Generators in S120
(1 43 21)(2 44 22)(3 45 23)(4 41 24)(5 42 25)(6 48 29)(7 49 30)(8 50 26)(9 46 27)(10 47 28)(11 55 33)(12 51 34)(13 52 35)(14 53 31)(15 54 32)(16 60 36)(17 56 37)(18 57 38)(19 58 39)(20 59 40)(61 107 87)(62 108 88)(63 109 89)(64 110 90)(65 101 81)(66 102 82)(67 103 83)(68 104 84)(69 105 85)(70 106 86)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 74 19 68 6 79 14 63)(2 80 20 64 7 75 15 69)(3 76 16 70 8 71 11 65)(4 72 17 66 9 77 12 61)(5 78 18 62 10 73 13 67)(21 94 39 84 29 99 31 89)(22 100 40 90 30 95 32 85)(23 96 36 86 26 91 33 81)(24 92 37 82 27 97 34 87)(25 98 38 88 28 93 35 83)(41 112 56 102 46 117 51 107)(42 118 57 108 47 113 52 103)(43 114 58 104 48 119 53 109)(44 120 59 110 49 115 54 105)(45 116 60 106 50 111 55 101)
(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)
(2 5)(3 4)(7 10)(8 9)(11 17)(12 16)(13 20)(14 19)(15 18)(22 25)(23 24)(26 27)(28 30)(31 39)(32 38)(33 37)(34 36)(35 40)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)(81 92)(82 91)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(101 112)(102 111)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)

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

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

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

93 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C5A5B6A6B6C6D6E···6J8A8B10A10B10C10D12A12B12C12D12E12F15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order122222334445566666···688101010101212121212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size11220202011222022112220···204422442222202022222222444444222244444···42···244444···4

93 irreducible representations

dim111111111111222222222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D5D10D10C3×D4C3×D4C3×D5D20D20C6×D5C6×D5C3×D20C3×D20C8⋊C22C3×C8⋊C22C8⋊D10C3×C8⋊D10
kernelC3×C8⋊D10C3×C40⋊C2C3×D40C15×M4(2)C6×D20C3×C4○D20C8⋊D10C40⋊C2D40C5×M4(2)C2×D20C4○D20C60C2×C30C3×M4(2)C24C2×C12C20C2×C10M4(2)C12C2×C6C8C2×C4C4C22C15C5C3C1
# reps122111244222112422244484881248

Matrix representation of C3×C8⋊D10 in GL6(𝔽241)

1500000
0150000
001000
000100
000010
000001
,
441560000
881970000
000010
000001
001978500
001534400
,
190520000
19000000
005118900
0051000
000019052
00001900
,
12400000
02400000
00124000
00024000
0000197200
000015344

G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[44,88,0,0,0,0,156,197,0,0,0,0,0,0,0,0,197,153,0,0,0,0,85,44,0,0,1,0,0,0,0,0,0,1,0,0],[190,190,0,0,0,0,52,0,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,190,190,0,0,0,0,52,0],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240,0,0,0,0,0,0,197,153,0,0,0,0,200,44] >;

C3×C8⋊D10 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_{10}
% in TeX

G:=Group("C3xC8:D10");
// GroupNames label

G:=SmallGroup(480,701);
// by ID

G=gap.SmallGroup(480,701);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,142,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^10=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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