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G = D1210D10order 480 = 25·3·5

4th semidirect product of D12 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.8D6, D1210D10, C60.13C23, Dic304C22, C3⋊C87D10, D4⋊S34D5, (D4×D5)⋊2S3, C15⋊D83C2, (C5×D4).5D6, (C4×D5).7D6, D4.9(S3×D5), C36(D8⋊D5), D4.D155C2, (C6×D5).62D4, C6.142(D4×D5), D125D51C2, C1516(C8⋊C22), C153C87C22, (C3×D4).20D10, C30.175(C2×D4), C6.D203C2, C52(D126C22), (C5×D12)⋊4C22, C20.32D63C2, C20.13(C22×S3), (C3×Dic5).13D4, (D5×C12).5C22, (C3×D20).4C22, (D4×C15).7C22, C12.13(C22×D5), D10.28(C3⋊D4), Dic5.22(C3⋊D4), (C3×D4×D5)⋊2C2, C4.13(C2×S3×D5), (C5×D4⋊S3)⋊5C2, (C5×C3⋊C8)⋊7C22, C2.24(D5×C3⋊D4), C10.45(C2×C3⋊D4), SmallGroup(480,565)

Series: Derived Chief Lower central Upper central

C1C60 — D1210D10
C1C5C15C30C60D5×C12D125D5 — D1210D10
C15C30C60 — D1210D10
C1C2C4D4

Generators and relations for D1210D10
 G = < a,b,c,d | a12=b2=c10=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >

Subgroups: 748 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3, C6, C6 [×3], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4 [×2], C22×C6, C5×S3, C3×D5 [×2], C30, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C4.Dic3, D4⋊S3, D4⋊S3, D4.S3 [×2], C4○D12, C6×D4, C3×Dic5, Dic15, C60, C6×D5, C6×D5 [×3], S3×C10, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D126C22, C5×C3⋊C8, C153C8, S3×Dic5, C15⋊D4, D5×C12, C3×D20, C3×C5⋊D4, C5×D12, Dic30, D4×C15, D5×C2×C6, D8⋊D5, C20.32D6, C15⋊D8, C6.D20, C5×D4⋊S3, D4.D15, D125D5, C3×D4×D5, D1210D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D126C22, C2×S3×D5, D8⋊D5, D5×C3⋊D4, D1210D10

Smallest permutation representation of D1210D10
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 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 96)(9 95)(10 94)(11 93)(12 92)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 120)(21 119)(22 118)(23 117)(24 116)(25 37)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(72 84)
(1 98 23 78 27 4 107 14 75 30)(2 105 24 73 28 11 108 21 76 25)(3 100 13 80 29 6 97 16 77 32)(5 102 15 82 31 8 99 18 79 34)(7 104 17 84 33 10 101 20 81 36)(9 106 19 74 35 12 103 22 83 26)(37 87 52 113 71 43 93 58 119 65)(38 94 53 120 72)(39 89 54 115 61 45 95 60 109 67)(40 96 55 110 62)(41 91 56 117 63 47 85 50 111 69)(42 86 57 112 64)(44 88 59 114 66)(46 90 49 116 68)(48 92 51 118 70)
(1 30)(2 25)(3 32)(4 27)(5 34)(6 29)(7 36)(8 31)(9 26)(10 33)(11 28)(12 35)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)(37 93)(38 88)(39 95)(40 90)(41 85)(42 92)(43 87)(44 94)(45 89)(46 96)(47 91)(48 86)(49 62)(50 69)(51 64)(52 71)(53 66)(54 61)(55 68)(56 63)(57 70)(58 65)(59 72)(60 67)(73 108)(74 103)(75 98)(76 105)(77 100)(78 107)(79 102)(80 97)(81 104)(82 99)(83 106)(84 101)(110 116)(112 118)(114 120)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D6E6F6G8A8B10A10B10C10D10E10F12A12B15A15B20A20B30A30B30C30D30E30F40A40B40C40D60A60B
order122222344455666666688101010101010121215152020303030303030404040406060
size114101220221060222441010202012602288242442044444488881212121288

45 irreducible representations

dim1111111122222222222244444448
type++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C3⋊D4C3⋊D4C8⋊C22S3×D5D4×D5D126C22C2×S3×D5D8⋊D5D5×C3⋊D4D1210D10
kernelD1210D10C20.32D6C15⋊D8C6.D20C5×D4⋊S3D4.D15D125D5C3×D4×D5D4×D5C3×Dic5C6×D5D4⋊S3C4×D5D20C5×D4C3⋊C8D12C3×D4Dic5D10C15D4C6C5C4C3C2C1
# reps1111111111121112222212222442

Matrix representation of D1210D10 in GL6(𝔽241)

1500000
2392250000
000010
000001
00240000
00024000
,
1312230000
1501100000
0011357128184
0018412857113
00128184128184
005711357113
,
24000000
02400000
0000190190
000051240
0019019000
005124000
,
100000
010000
0000190190
000024051
0019019000
002405100

G:=sub<GL(6,GF(241))| [15,239,0,0,0,0,0,225,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[131,150,0,0,0,0,223,110,0,0,0,0,0,0,113,184,128,57,0,0,57,128,184,113,0,0,128,57,128,57,0,0,184,113,184,113],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,190,51,0,0,0,0,190,240,0,0,190,51,0,0,0,0,190,240,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,190,240,0,0,0,0,190,51,0,0,190,240,0,0,0,0,190,51,0,0] >;

D1210D10 in GAP, Magma, Sage, TeX

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

G:=Group("D12:10D10");
// GroupNames label

G:=SmallGroup(480,565);
// by ID

G=gap.SmallGroup(480,565);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,135,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

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