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G = D125D10order 480 = 25·3·5

5th semidirect product of D12 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D125D10, Dic105D6, D30.40D4, C60.24C23, D60.7C22, Dic15.14D4, D4⋊S36D5, C3⋊C810D10, (D4×D15)⋊3C2, C52C810D6, C5⋊D244C2, C53(Q83D6), D4.D56S3, C6.74(D4×D5), C34(D8⋊D5), D4.18(S3×D5), (C5×D4).12D6, C10.75(S3×D4), D12⋊D52C2, C1521(C8⋊C22), (C3×D4).12D10, C30.186(C2×D4), C15⋊SD164C2, (C5×D12)⋊5C22, D30.5C44C2, C20.24(C22×S3), (C4×D15).8C22, C12.24(C22×D5), (C3×Dic10)⋊5C22, (D4×C15).18C22, C2.27(D10⋊D6), C4.24(C2×S3×D5), (C5×D4⋊S3)⋊8C2, (C5×C3⋊C8)⋊10C22, (C3×D4.D5)⋊8C2, (C3×C52C8)⋊10C22, SmallGroup(480,576)

Series: Derived Chief Lower central Upper central

C1C60 — D125D10
C1C5C15C30C60C3×Dic10D12⋊D5 — D125D10
C15C30C60 — D125D10
C1C2C4D4

Generators and relations for D125D10
 G = < a,b,c,d | a12=b2=c10=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a9b, dbd=a7b, dcd=c-1 >

Subgroups: 988 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, C4×S3 [×2], D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, D15 [×2], C30, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C3×Dic5, Dic15, C60, S3×C10, D30, D30 [×3], C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, Q83D6, C5×C3⋊C8, C3×C52C8, S3×Dic5, C5⋊D12, C3×Dic10, C5×D12, C4×D15, D60, C157D4, D4×C15, C22×D15, D8⋊D5, D30.5C4, C5⋊D24, C15⋊SD16, C3×D4.D5, C5×D4⋊S3, D12⋊D5, D4×D15, D125D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, Q83D6, C2×S3×D5, D8⋊D5, D10⋊D6, D125D10

Smallest permutation representation of D125D10
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)
(1 111)(2 110)(3 109)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 97)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 101)(22 100)(23 99)(24 98)(25 40)(26 39)(27 38)(28 37)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 84)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 96)(69 95)(70 94)(71 93)(72 92)
(1 42 52 96 17 4 39 55 93 20)(2 37 53 91 18 11 40 50 94 15)(3 44 54 86 19 6 41 57 95 22)(5 46 56 88 21 8 43 59 85 24)(7 48 58 90 23 10 45 49 87 14)(9 38 60 92 13 12 47 51 89 16)(25 73 70 98 110 31 79 64 104 116)(26 80 71 105 111)(27 75 72 100 112 33 81 66 106 118)(28 82 61 107 113)(29 77 62 102 114 35 83 68 108 120)(30 84 63 97 115)(32 74 65 99 117)(34 76 67 101 119)(36 78 69 103 109)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 24)(10 23)(11 22)(12 21)(25 62)(26 61)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 86)(38 85)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(73 77)(74 76)(78 84)(79 83)(80 82)(97 109)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D10E10F12A12B15A15B20A20B24A24B30A30B30C30D30E30F40A40B40C40D60A60B
order12222234445566881010101010101212151520202424303030303030404040406060
size1141230602220302228122022882424440444420204488881212121288

42 irreducible representations

dim111111112222222222444444448
type++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C8⋊C22S3×D4S3×D5D4×D5Q83D6C2×S3×D5D8⋊D5D10⋊D6D125D10
kernelD125D10D30.5C4C5⋊D24C15⋊SD16C3×D4.D5C5×D4⋊S3D12⋊D5D4×D15D4.D5Dic15D30D4⋊S3C52C8Dic10C5×D4C3⋊C8D12C3×D4C15C10D4C6C5C4C3C2C1
# reps111111111112111222112222442

Matrix representation of D125D10 in GL8(𝔽241)

2400000000
0240000000
0024000000
0002400000
00002402401771
00002402401772
00001772262192
00000100
,
10200000
01020000
0024000000
0002400000
00002322321940
0000116947107
0000000200
000000470
,
5151000000
1901000000
1901901901900000
51240512400000
00000001
00001164240
0000002400
00001000
,
5151000000
1190000000
1901901901900000
24051240510000
00001164240
00000001
0000002400
00000100

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,240,177,0,0,0,0,0,240,240,226,1,0,0,0,0,177,177,2,0,0,0,0,0,1,2,192,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,240,0,0,0,0,0,0,2,0,240,0,0,0,0,0,0,0,0,232,116,0,0,0,0,0,0,232,9,0,0,0,0,0,0,194,47,0,47,0,0,0,0,0,107,200,0],[51,190,190,51,0,0,0,0,51,1,190,240,0,0,0,0,0,0,190,51,0,0,0,0,0,0,190,240,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,64,240,0,0,0,0,0,1,240,0,0],[51,1,190,240,0,0,0,0,51,190,190,51,0,0,0,0,0,0,190,240,0,0,0,0,0,0,190,51,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,64,0,240,0,0,0,0,0,240,1,0,0] >;

D125D10 in GAP, Magma, Sage, TeX

D_{12}\rtimes_5D_{10}
% in TeX

G:=Group("D12:5D10");
// GroupNames label

G:=SmallGroup(480,576);
// by ID

G=gap.SmallGroup(480,576);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,303,675,346,185,80,1356,18822]);
// 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^7,c*b*c^-1=a^9*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations

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