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G = D12⋊D10order 480 = 25·3·5

6th semidirect product of D12 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D126D10, D20.26D6, C60.28C23, D60.11C22, Q8⋊D51S3, Q83(S3×D5), (S3×D20)⋊3C2, C5⋊D245C2, C52C811D6, C15⋊D86C2, C57(Q83D6), (C3×Q8)⋊2D10, (C5×Q8)⋊11D6, Q83S31D5, (C4×S3).9D10, C33(D4⋊D10), C1523(C8⋊C22), Q82D154C2, (S3×C10).12D4, C10.148(S3×D4), C30.190(C2×D4), D6.8(C5⋊D4), (C5×D12)⋊6C22, D6.Dic55C2, (Q8×C15)⋊4C22, C153C814C22, C20.28(C22×S3), (C5×Dic3).38D4, (C3×D20).9C22, C12.28(C22×D5), (S3×C20).10C22, Dic3.17(C5⋊D4), C4.28(C2×S3×D5), (C3×Q8⋊D5)⋊2C2, C2.29(S3×C5⋊D4), C6.51(C2×C5⋊D4), (C5×Q83S3)⋊1C2, (C3×C52C8)⋊12C22, SmallGroup(480,580)

Series: Derived Chief Lower central Upper central

C1C60 — D12⋊D10
C1C5C15C30C60C3×D20S3×D20 — D12⋊D10
C15C30C60 — D12⋊D10
C1C2C4Q8

Generators and relations for D12⋊D10
 G = < a,b,c,d | a12=b2=c10=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a7b, dcd=c-1 >

Subgroups: 892 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6, D6 [×4], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20, C20 [×2], D10 [×4], C2×C10 [×2], C3⋊C8, C24, C4×S3, C4×S3, D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C3×D5, D15, C30, C8⋊C22, C52C8, C52C8, D20, D20 [×2], C2×C20 [×2], C5×D4 [×2], C5×Q8, C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C5×Dic3, C60, C60, S3×D5 [×2], C6×D5, S3×C10, S3×C10, D30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, Q83D6, C3×C52C8, C153C8, C3⋊D20, C3×D20, S3×C20, S3×C20, C5×D12, C5×D12, D60, Q8×C15, C2×S3×D5, D4⋊D10, D6.Dic5, C15⋊D8, C5⋊D24, C3×Q8⋊D5, Q82D15, S3×D20, C5×Q83S3, D12⋊D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q83D6, C2×S3×D5, D4⋊D10, S3×C5⋊D4, D12⋊D10

Smallest permutation representation of D12⋊D10
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 88)(2 87)(3 86)(4 85)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 99)(14 98)(15 97)(16 108)(17 107)(18 106)(19 105)(20 104)(21 103)(22 102)(23 101)(24 100)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 72)(34 71)(35 70)(36 69)(37 113)(38 112)(39 111)(40 110)(41 109)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 84)(60 83)
(1 56 31 46 108)(2 49 32 39 97 6 57 36 47 101)(3 54 33 44 98 11 58 29 48 106)(4 59 34 37 99)(5 52 35 42 100 9 60 27 38 104)(7 50 25 40 102)(8 55 26 45 103 12 51 30 41 107)(10 53 28 43 105)(13 87 84 61 113 15 85 74 71 115)(14 92 73 66 114 20 86 79 72 120)(16 90 75 64 116 18 88 77 62 118)(17 95 76 69 117 23 89 82 63 111)(19 93 78 67 119 21 91 80 65 109)(22 96 81 70 110 24 94 83 68 112)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 90)(20 89)(21 88)(22 87)(23 86)(24 85)(26 36)(27 35)(28 34)(29 33)(30 32)(37 53)(38 52)(39 51)(40 50)(41 49)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 111)(74 110)(75 109)(76 120)(77 119)(78 118)(79 117)(80 116)(81 115)(82 114)(83 113)(84 112)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C···10H12A12B15A15B20A···20F20G20H20I20J24A24B30A30B60A···60F
order1222223444556688101010···101212151520···20202020202424303060···60
size11612206022462224020602212···1248444···466662020448···8

48 irreducible representations

dim1111111122222222222244444448
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C5⋊D4C5⋊D4C8⋊C22S3×D4S3×D5Q83D6C2×S3×D5D4⋊D10S3×C5⋊D4D12⋊D10
kernelD12⋊D10D6.Dic5C15⋊D8C5⋊D24C3×Q8⋊D5Q82D15S3×D20C5×Q83S3Q8⋊D5C5×Dic3S3×C10Q83S3C52C8D20C5×Q8C4×S3D12C3×Q8Dic3D6C15C10Q8C5C4C3C2C1
# reps1111111111121112224411222442

Matrix representation of D12⋊D10 in GL6(𝔽241)

100000
010000
0024024022
00102390
0024024011
00102400
,
24000000
02400000
00002329
0000189
0011612500
00912500
,
52520000
1892400000
001000
0024024000
000010
0000240240
,
52520000
2401890000
001000
0024024000
00102400
0024024011

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,240,1,0,0,240,0,240,0,0,0,2,239,1,240,0,0,2,0,1,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,116,9,0,0,0,0,125,125,0,0,232,18,0,0,0,0,9,9,0,0],[52,189,0,0,0,0,52,240,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,240],[52,240,0,0,0,0,52,189,0,0,0,0,0,0,1,240,1,240,0,0,0,240,0,240,0,0,0,0,240,1,0,0,0,0,0,1] >;

D12⋊D10 in GAP, Magma, Sage, TeX

D_{12}\rtimes D_{10}
% in TeX

G:=Group("D12:D10");
// GroupNames label

G:=SmallGroup(480,580);
// by ID

G=gap.SmallGroup(480,580);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,100,675,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations

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