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G = C10×D12order 240 = 24·3·5

Direct product of C10 and D12

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

Aliases: C10×D12, C209D6, C306D4, C6011C22, C30.50C23, C61(C5×D4), C31(D4×C10), C42(S3×C10), (C2×C20)⋊6S3, C1512(C2×D4), (C2×C60)⋊10C2, C122(C2×C10), (C2×C12)⋊3C10, D61(C2×C10), (C2×C10).38D6, (C22×S3)⋊1C10, (S3×C10)⋊9C22, C6.3(C22×C10), C10.40(C22×S3), C22.10(S3×C10), (C2×C30).49C22, (S3×C2×C10)⋊5C2, (C2×C4)⋊2(C5×S3), C2.4(S3×C2×C10), (C2×C6).10(C2×C10), SmallGroup(240,167)

Series: Derived Chief Lower central Upper central

C1C6 — C10×D12
C1C3C6C30S3×C10S3×C2×C10 — C10×D12
C3C6 — C10×D12
C1C2×C10C2×C20

Generators and relations for C10×D12
 G = < a,b,c | a10=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 248 in 108 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], C10, C10 [×2], C10 [×4], C12 [×2], D6 [×4], D6 [×4], C2×C6, C15, C2×D4, C20 [×2], C2×C10, C2×C10 [×8], D12 [×4], C2×C12, C22×S3 [×2], C5×S3 [×4], C30, C30 [×2], C2×C20, C5×D4 [×4], C22×C10 [×2], C2×D12, C60 [×2], S3×C10 [×4], S3×C10 [×4], C2×C30, D4×C10, C5×D12 [×4], C2×C60, S3×C2×C10 [×2], C10×D12
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, C5×D4 [×2], C22×C10, C2×D12, S3×C10 [×3], D4×C10, C5×D12 [×2], S3×C2×C10, C10×D12

Smallest permutation representation of C10×D12
On 120 points
Generators in S120
(1 28 72 81 15 102 56 40 117 89)(2 29 61 82 16 103 57 41 118 90)(3 30 62 83 17 104 58 42 119 91)(4 31 63 84 18 105 59 43 120 92)(5 32 64 73 19 106 60 44 109 93)(6 33 65 74 20 107 49 45 110 94)(7 34 66 75 21 108 50 46 111 95)(8 35 67 76 22 97 51 47 112 96)(9 36 68 77 23 98 52 48 113 85)(10 25 69 78 24 99 53 37 114 86)(11 26 70 79 13 100 54 38 115 87)(12 27 71 80 14 101 55 39 116 88)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 76)(74 75)(77 84)(78 83)(79 82)(80 81)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)(97 106)(98 105)(99 104)(100 103)(101 102)(107 108)(109 112)(110 111)(113 120)(114 119)(115 118)(116 117)

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

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

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

C10×D12 is a maximal subgroup of
C20.5D12  D12⋊Dic5  C10.D24  D2021D6  D6036C22  C60.89D4  C60.69D4  Dic5⋊D12  (C2×D12).D5  Dic158D4  C60⋊D4  C20⋊D12  Dic152D4  C6010D4  C202D12  D304D4  D2026D6  S3×D4×C10

90 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B5C5D6A6B6C10A···10L10M···10AB12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order12222222344555566610···1010···10121212121515151520···2030···3060···60
size1111666622211112221···16···6222222222···22···22···2

90 irreducible representations

dim111111112222222222
type+++++++++
imageC1C2C2C2C5C10C10C10S3D4D6D6D12C5×S3C5×D4S3×C10S3×C10C5×D12
kernelC10×D12C5×D12C2×C60S3×C2×C10C2×D12D12C2×C12C22×S3C2×C20C30C20C2×C10C10C2×C4C6C4C22C2
# reps14124164812214488416

Matrix representation of C10×D12 in GL4(𝔽61) generated by

27000
02700
0090
0009
,
1100
60000
002338
002346
,
1100
06000
003823
004623
G:=sub<GL(4,GF(61))| [27,0,0,0,0,27,0,0,0,0,9,0,0,0,0,9],[1,60,0,0,1,0,0,0,0,0,23,23,0,0,38,46],[1,0,0,0,1,60,0,0,0,0,38,46,0,0,23,23] >;

C10×D12 in GAP, Magma, Sage, TeX

C_{10}\times D_{12}
% in TeX

G:=Group("C10xD12");
// GroupNames label

G:=SmallGroup(240,167);
// by ID

G=gap.SmallGroup(240,167);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,794,194,5765]);
// Polycyclic

G:=Group<a,b,c|a^10=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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