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G = C5×D126C22order 480 = 25·3·5

Direct product of C5 and D126C22

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×D126C22, C60.148D4, C60.228C23, D4⋊S35C10, (C6×D4)⋊2C10, (D4×C10)⋊9S3, C4○D123C10, (D4×C30)⋊16C2, D126(C2×C10), D4.S35C10, (C5×D4).35D6, D4.6(S3×C10), C12.15(C5×D4), C6.45(D4×C10), C1536(C8⋊C22), Dic65(C2×C10), (C2×C30).182D4, C30.428(C2×D4), (C2×C20).242D6, C4.Dic36C10, (C5×D12)⋊36C22, C20.95(C3⋊D4), C20.201(C22×S3), (C2×C60).358C22, C12.12(C22×C10), (C5×Dic6)⋊32C22, (D4×C15).45C22, C3⋊C83(C2×C10), C34(C5×C8⋊C22), (C2×D4)⋊2(C5×S3), C4.12(S3×C2×C10), (C5×D4⋊S3)⋊13C2, (C5×C3⋊C8)⋊25C22, (C2×C6).39(C5×D4), C4.16(C5×C3⋊D4), C2.9(C10×C3⋊D4), (C5×C4○D12)⋊13C2, (C2×C4).15(S3×C10), (C5×D4.S3)⋊13C2, (C3×D4).6(C2×C10), (C2×C12).31(C2×C10), C10.130(C2×C3⋊D4), (C5×C4.Dic3)⋊18C2, C22.10(C5×C3⋊D4), (C2×C10).63(C3⋊D4), SmallGroup(480,811)

Series: Derived Chief Lower central Upper central

C1C12 — C5×D126C22
C1C3C6C12C60C5×D12C5×C4○D12 — C5×D126C22
C3C6C12 — C5×D126C22
C1C10C2×C20D4×C10

Generators and relations for C5×D126C22
 G = < a,b,c,d,e | a5=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >

Subgroups: 324 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, C10, C10 [×4], Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, C2×C10, C2×C10 [×5], C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C5×S3, C30, C30 [×3], C8⋊C22, C40 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×D4 [×3], C5×Q8, C22×C10, C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C4○D12, C6×D4, C5×Dic3, C60 [×2], S3×C10, C2×C30, C2×C30 [×4], C5×M4(2), C5×D8 [×2], C5×SD16 [×2], D4×C10, C5×C4○D4, D126C22, C5×C3⋊C8 [×2], C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4×C15 [×2], D4×C15, C22×C30, C5×C8⋊C22, C5×C4.Dic3, C5×D4⋊S3 [×2], C5×D4.S3 [×2], C5×C4○D12, D4×C30, C5×D126C22
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C8⋊C22, C5×D4 [×2], C22×C10, C2×C3⋊D4, S3×C10 [×3], D4×C10, D126C22, C5×C3⋊D4 [×2], S3×C2×C10, C5×C8⋊C22, C10×C3⋊D4, C5×D126C22

Smallest permutation representation of C5×D126C22
On 120 points
Generators in S120
(1 59 47 30 20)(2 60 48 31 21)(3 49 37 32 22)(4 50 38 33 23)(5 51 39 34 24)(6 52 40 35 13)(7 53 41 36 14)(8 54 42 25 15)(9 55 43 26 16)(10 56 44 27 17)(11 57 45 28 18)(12 58 46 29 19)(61 118 97 85 76)(62 119 98 86 77)(63 120 99 87 78)(64 109 100 88 79)(65 110 101 89 80)(66 111 102 90 81)(67 112 103 91 82)(68 113 104 92 83)(69 114 105 93 84)(70 115 106 94 73)(71 116 107 95 74)(72 117 108 96 75)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 72)(11 71)(12 70)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 84)(21 83)(22 82)(23 81)(24 80)(25 86)(26 85)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(37 103)(38 102)(39 101)(40 100)(41 99)(42 98)(43 97)(44 108)(45 107)(46 106)(47 105)(48 104)(49 112)(50 111)(51 110)(52 109)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)
(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(1 4)(2 11)(3 6)(5 8)(7 10)(9 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)(25 34)(26 29)(27 36)(28 31)(30 33)(32 35)(37 40)(38 47)(39 42)(41 44)(43 46)(45 48)(49 52)(50 59)(51 54)(53 56)(55 58)(57 60)(61 67)(63 69)(65 71)(74 80)(76 82)(78 84)(85 91)(87 93)(89 95)(97 103)(99 105)(101 107)(110 116)(112 118)(114 120)

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

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

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

105 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B5C5D6A6B6C6D6E6F6G8A8B10A10B10C10D10E10F10G10H10I···10P10Q10R10S10T12A12B15A15B15C15D20A···20H20I20J20K20L30A···30L30M···30AB40A···40H60A···60H
order12222234445555666666688101010101010101010···101010101012121515151520···202020202030···3030···3040···4060···60
size112441222212111122244441212111122224···4121212124422222···2121212122···24···412···124···4

105 irreducible representations

dim111111111111222222222222224444
type++++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D4D6D6C3⋊D4C3⋊D4C5×S3C5×D4C5×D4S3×C10S3×C10C5×C3⋊D4C5×C3⋊D4C8⋊C22D126C22C5×C8⋊C22C5×D126C22
kernelC5×D126C22C5×C4.Dic3C5×D4⋊S3C5×D4.S3C5×C4○D12D4×C30D126C22C4.Dic3D4⋊S3D4.S3C4○D12C6×D4D4×C10C60C2×C30C2×C20C5×D4C20C2×C10C2×D4C12C2×C6C2×C4D4C4C22C15C5C3C1
# reps112211448844111122244448881248

Matrix representation of C5×D126C22 in GL4(𝔽241) generated by

91000
09100
00910
00091
,
022500
16000
000226
00150
,
000226
00150
022500
16000
,
1000
0100
002400
000240
,
0100
1000
0010
000240
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[0,16,0,0,225,0,0,0,0,0,0,15,0,0,226,0],[0,0,0,16,0,0,225,0,0,15,0,0,226,0,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,240] >;

C5×D126C22 in GAP, Magma, Sage, TeX

C_5\times D_{12}\rtimes_6C_2^2
% in TeX

G:=Group("C5xD12:6C2^2");
// GroupNames label

G:=SmallGroup(480,811);
// by ID

G=gap.SmallGroup(480,811);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,4204,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^12=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations

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