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G = D1214D10order 480 = 25·3·5

8th semidirect product of D12 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1214D10, Dic1013D6, D6010C22, C30.31C24, C60.55C23, C1562+ 1+4, D30.16C23, Dic15.18C23, (C4×D5)⋊5D6, (S3×D4)⋊5D5, C5⋊D47D6, (D4×D15)⋊7C2, (D5×D12)⋊5C2, (C4×S3)⋊5D10, D412(S3×D5), (C5×D4)⋊15D6, C53(D4○D12), C3⋊D47D10, (C3×D4)⋊15D10, D42D55S3, (C2×Dic5)⋊8D6, C15⋊Q814C22, D12⋊D55C2, D10⋊D65C2, D60⋊C25C2, C33(D46D10), (S3×C20)⋊6C22, (C4×D15)⋊6C22, (D5×C12)⋊6C22, (C22×S3)⋊6D10, C157D47C22, (C2×C30).7C23, C6.31(C23×D5), D6.D104C2, Dic3.D105C2, (D4×C15)⋊13C22, (C5×D12)⋊10C22, C5⋊D1216C22, C3⋊D2015C22, C15⋊D416C22, C20.55(C22×S3), C10.31(S3×C23), D30.C23C22, (S3×Dic5)⋊3C22, (C6×D5).13C23, D6.27(C22×D5), C12.55(C22×D5), (S3×C10).16C23, (C6×Dic5)⋊14C22, (C3×Dic10)⋊9C22, D10.16(C22×S3), (C22×D15)⋊13C22, Dic5.17(C22×S3), Dic3.16(C22×D5), (C3×Dic5).46C23, (C5×Dic3).17C23, (C5×S3×D4)⋊7C2, C4.55(C2×S3×D5), (S3×C5⋊D4)⋊5C2, (C2×S3×D5)⋊6C22, C22.7(C2×S3×D5), (S3×C2×C10)⋊9C22, (C3×D42D5)⋊7C2, (C2×C5⋊D12)⋊20C2, C2.34(C22×S3×D5), (C5×C3⋊D4)⋊7C22, (C3×C5⋊D4)⋊7C22, (C2×C6).7(C22×D5), (C2×C10).7(C22×S3), SmallGroup(480,1103)

Series: Derived Chief Lower central Upper central

C1C30 — D1214D10
C1C5C15C30C6×D5C2×S3×D5D5×D12 — D1214D10
C15C30 — D1214D10
C1C2D4

Generators and relations for D1214D10
 G = < a,b,c,d | a12=b2=c10=d2=1, bab=a-1, cac-1=a7, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >

Subgroups: 1884 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×5], C22 [×2], C22 [×13], C5, S3 [×6], C6, C6 [×3], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], D5 [×4], C10, C10 [×5], Dic3, Dic3, C12, C12 [×3], D6, D6 [×2], D6 [×9], C2×C6 [×2], C2×C6, C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5 [×2], Dic5, C20, C20, D10, D10 [×7], C2×C10 [×2], C2×C10 [×5], Dic6, C4×S3, C4×S3 [×5], D12, D12 [×8], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C22×S3 [×2], C22×S3 [×4], C5×S3 [×3], C3×D5, D15 [×3], C30, C30 [×2], 2+ 1+4, Dic10, Dic10, C4×D5, C4×D5 [×3], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4, C5×D4 [×3], C22×D5 [×4], C22×C10 [×2], C2×D12 [×3], C4○D12 [×3], S3×D4, S3×D4 [×5], Q83S3 [×2], C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5 [×2], Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, S3×C10 [×2], S3×C10 [×2], D30, D30 [×2], D30 [×2], C2×C30 [×2], C4○D20 [×2], D4×D5 [×4], D42D5, D42D5 [×3], C2×C5⋊D4 [×4], D4×C10, D4○D12, S3×Dic5 [×2], D30.C2 [×2], C15⋊D4, C3⋊D20, C5⋊D12, C5⋊D12 [×6], C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], S3×C20, C5×D12, C5×C3⋊D4 [×2], C4×D15, D60, C157D4 [×2], D4×C15, C2×S3×D5 [×2], S3×C2×C10 [×2], C22×D15 [×2], D46D10, D12⋊D5, D60⋊C2, D6.D10, D5×D12, Dic3.D10 [×2], C2×C5⋊D12 [×2], S3×C5⋊D4 [×2], D10⋊D6 [×2], C3×D42D5, C5×S3×D4, D4×D15, D1214D10
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5 [×3], D46D10, C22×S3×D5, D1214D10

Smallest permutation representation of D1214D10
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 82)(74 81)(75 80)(76 79)(77 78)(83 84)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 102)(98 101)(99 100)(103 108)(104 107)(105 106)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)
(1 19 41 58 91)(2 14 42 53 92 8 20 48 59 86)(3 21 43 60 93)(4 16 44 55 94 10 22 38 49 88)(5 23 45 50 95)(6 18 46 57 96 12 24 40 51 90)(7 13 47 52 85)(9 15 37 54 87)(11 17 39 56 89)(25 102 80 61 115 31 108 74 67 109)(26 97 81 68 116)(27 104 82 63 117 33 98 76 69 111)(28 99 83 70 118)(29 106 84 65 119 35 100 78 71 113)(30 101 73 72 120)(32 103 75 62 110)(34 105 77 64 112)(36 107 79 66 114)
(1 75)(2 80)(3 73)(4 78)(5 83)(6 76)(7 81)(8 74)(9 79)(10 84)(11 77)(12 82)(13 97)(14 102)(15 107)(16 100)(17 105)(18 98)(19 103)(20 108)(21 101)(22 106)(23 99)(24 104)(25 42)(26 47)(27 40)(28 45)(29 38)(30 43)(31 48)(32 41)(33 46)(34 39)(35 44)(36 37)(49 113)(50 118)(51 111)(52 116)(53 109)(54 114)(55 119)(56 112)(57 117)(58 110)(59 115)(60 120)(61 86)(62 91)(63 96)(64 89)(65 94)(66 87)(67 92)(68 85)(69 90)(70 95)(71 88)(72 93)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D10A10B10C10D10E10F10G10H10I10J10K10L10M10N12A12B12C12D12E15A15B20A20B20C20D30A30B30C30D30E30F60A60B
order122222222223444444556666101010101010101010101010101012121212121515202020203030303030306060
size1122666103030302261010103022244202244446666121212124101020204444121244888888

57 irreducible representations

dim1111111111112222222222224444448
type++++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6D10D10D10D10D102+ 1+4S3×D5D4○D12C2×S3×D5C2×S3×D5D46D10D1214D10
kernelD1214D10D12⋊D5D60⋊C2D6.D10D5×D12Dic3.D10C2×C5⋊D12S3×C5⋊D4D10⋊D6C3×D42D5C5×S3×D4D4×D15D42D5S3×D4Dic10C4×D5C2×Dic5C5⋊D4C5×D4C4×S3D12C3⋊D4C3×D4C22×S3C15D4C5C4C22C3C1
# reps1111122221111211221224241222442

Matrix representation of D1214D10 in GL8(𝔽61)

10100000
01010000
600000000
060000000
00000100
000060000
00000001
000000600
,
10100000
01010000
006000000
000600000
00000100
00001000
00000001
00000010
,
6044000000
1744000000
0060440000
0017440000
00009000
000005200
000000270
000000034
,
3047000000
2531000000
311431140000
363036300000
000000027
000000340
00000900
000052000

G:=sub<GL(8,GF(61))| [1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[60,17,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,60,17,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,34],[30,25,31,36,0,0,0,0,47,31,14,30,0,0,0,0,0,0,31,36,0,0,0,0,0,0,14,30,0,0,0,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,9,0,0,0,0,0,0,34,0,0,0,0,0,0,27,0,0,0] >;

D1214D10 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{14}D_{10}
% in TeX

G:=Group("D12:14D10");
// GroupNames label

G:=SmallGroup(480,1103);
// by ID

G=gap.SmallGroup(480,1103);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,185,1356,18822]);
// 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=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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