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G = D12⋊D5order 240 = 24·3·5

3rd semidirect product of D12 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D123D5, D6.1D10, C20.15D6, Dic103S3, C12.16D10, C30.5C23, Dic5.2D6, C60.27C22, D30.9C22, Dic15.11C22, (C4×D15)⋊6C2, (C5×D12)⋊5C2, C153(C4○D4), C5⋊D121C2, C4.20(S3×D5), C31(D42D5), (S3×Dic5)⋊1C2, C52(Q83S3), C6.5(C22×D5), (C3×Dic10)⋊5C2, C10.5(C22×S3), (S3×C10).1C22, (C3×Dic5).2C22, C2.9(C2×S3×D5), SmallGroup(240,129)

Series: Derived Chief Lower central Upper central

C1C30 — D12⋊D5
C1C5C15C30C3×Dic5S3×Dic5 — D12⋊D5
C15C30 — D12⋊D5
C1C2C4

Generators and relations for D12⋊D5
 G = < a,b,c,d | a12=b2=c5=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 344 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×3], C6, C2×C4 [×3], D4 [×3], Q8, D5, C10, C10 [×2], Dic3, C12, C12 [×2], D6 [×2], D6, C15, C4○D4, Dic5 [×2], Dic5, C20, D10, C2×C10 [×2], C4×S3 [×3], D12, D12 [×2], C3×Q8, C5×S3 [×2], D15, C30, Dic10, C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4, Q83S3, C3×Dic5 [×2], Dic15, C60, S3×C10 [×2], D30, D42D5, S3×Dic5 [×2], C5⋊D12 [×2], C3×Dic10, C5×D12, C4×D15, D12⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, Q83S3, S3×D5, D42D5, C2×S3×D5, D12⋊D5

Smallest permutation representation of D12⋊D5
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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 84)(20 83)(21 82)(22 81)(23 80)(24 79)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 53)(38 52)(39 51)(40 50)(41 49)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(97 111)(98 110)(99 109)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)
(1 42 93 77 97)(2 43 94 78 98)(3 44 95 79 99)(4 45 96 80 100)(5 46 85 81 101)(6 47 86 82 102)(7 48 87 83 103)(8 37 88 84 104)(9 38 89 73 105)(10 39 90 74 106)(11 40 91 75 107)(12 41 92 76 108)(13 110 71 59 27)(14 111 72 60 28)(15 112 61 49 29)(16 113 62 50 30)(17 114 63 51 31)(18 115 64 52 32)(19 116 65 53 33)(20 117 66 54 34)(21 118 67 55 35)(22 119 68 56 36)(23 120 69 57 25)(24 109 70 58 26)
(1 97)(2 102)(3 107)(4 100)(5 105)(6 98)(7 103)(8 108)(9 101)(10 106)(11 99)(12 104)(13 57)(14 50)(15 55)(16 60)(17 53)(18 58)(19 51)(20 56)(21 49)(22 54)(23 59)(24 52)(25 27)(26 32)(28 30)(29 35)(31 33)(34 36)(37 76)(38 81)(39 74)(40 79)(41 84)(42 77)(43 82)(44 75)(45 80)(46 73)(47 78)(48 83)(61 118)(62 111)(63 116)(64 109)(65 114)(66 119)(67 112)(68 117)(69 110)(70 115)(71 120)(72 113)(85 89)(86 94)(88 92)(91 95)

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

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

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

D12⋊D5 is a maximal subgroup of
D246D5  Dic6.D10  D245D5  D30.4D4  D30.11D4  D125D10  D12.D10  D30.44D4  C30.C24  D2024D6  D2026D6  S3×D42D5  D1214D10  D12.29D10  D5×Q83S3
D12⋊D5 is a maximal quotient of
(C2×C20).D6  Dic156Q8  Dic5.2Dic6  C4⋊Dic3⋊D5  D6⋊C4.D5  (C4×D15)⋊8C4  D30.35D4  D6⋊Dic5.C2  C60.89D4  C12.Dic10  D3010Q8  (S3×Dic5)⋊C4  Dic1514D4  D61Dic10  Dic5⋊D12  (C2×D12).D5  D6.D20  Dic158D4  C202D12

33 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B 6 10A10B10C10D10E10F12A12B12C15A15B20A20B30A30B60A60B60C60D
order1222234444455610101010101012121215152020303060606060
size11663022101015152222212121212420204444444444

33 irreducible representations

dim111111222222244444
type++++++++++++++-+
imageC1C2C2C2C2C2S3D5D6D6C4○D4D10D10Q83S3S3×D5D42D5C2×S3×D5D12⋊D5
kernelD12⋊D5S3×Dic5C5⋊D12C3×Dic10C5×D12C4×D15Dic10D12Dic5C20C15C12D6C5C4C3C2C1
# reps122111122122412224

Matrix representation of D12⋊D5 in GL6(𝔽61)

60600000
100000
001000
000100
0000500
00002111
,
110000
0600000
0060000
0006000
00002050
0000341
,
100000
010000
000100
00601700
000010
000001
,
100000
60600000
000100
001000
000010
00004860

G:=sub<GL(6,GF(61))| [60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,21,0,0,0,0,0,11],[1,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,20,3,0,0,0,0,50,41],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,48,0,0,0,0,0,60] >;

D12⋊D5 in GAP, Magma, Sage, TeX

D_{12}\rtimes D_5
% in TeX

G:=Group("D12:D5");
// GroupNames label

G:=SmallGroup(240,129);
// by ID

G=gap.SmallGroup(240,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,218,116,50,490,6917]);
// Polycyclic

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

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