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G = C5×D4⋊D6order 480 = 25·3·5

Direct product of C5 and D4⋊D6

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

Aliases: C5×D4⋊D6, C60.227D4, C60.233C23, D4⋊S36C10, D44(S3×C10), (C5×D4)⋊26D6, (C5×Q8)⋊26D6, Q85(S3×C10), C6.59(D4×C10), C12.49(C5×D4), (C2×C30).95D4, (C10×D12)⋊26C2, (C2×D12)⋊10C10, C1539(C8⋊C22), Q82S36C10, C30.442(C2×D4), (C2×C20).248D6, C4.Dic39C10, D12.11(C2×C10), (D4×C15)⋊35C22, (Q8×C15)⋊31C22, C20.117(C3⋊D4), C12.17(C22×C10), (C2×C60).370C22, C20.206(C22×S3), (C5×D12).50C22, C3⋊C84(C2×C10), C35(C5×C8⋊C22), C4○D43(C5×S3), C4.17(S3×C2×C10), (C2×C6).8(C5×D4), (C5×D4⋊S3)⋊14C2, (C5×C4○D4)⋊10S3, (C3×C4○D4)⋊1C10, (C3×D4)⋊4(C2×C10), (C5×C3⋊C8)⋊26C22, (C3×Q8)⋊4(C2×C10), C4.24(C5×C3⋊D4), (C15×C4○D4)⋊11C2, (C2×C4).20(S3×C10), C2.23(C10×C3⋊D4), C22.5(C5×C3⋊D4), (C2×C12).44(C2×C10), (C5×Q82S3)⋊14C2, C10.144(C2×C3⋊D4), (C5×C4.Dic3)⋊21C2, (C2×C10).41(C3⋊D4), SmallGroup(480,828)

Series: Derived Chief Lower central Upper central

C1C12 — C5×D4⋊D6
C1C3C6C12C60C5×D12C10×D12 — C5×D4⋊D6
C3C6C12 — C5×D4⋊D6
C1C10C2×C20C5×C4○D4

Generators and relations for C5×D4⋊D6
 G = < a,b,c,d,e | a5=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 372 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C10, C10 [×4], C12 [×2], C12, D6 [×4], C2×C6, C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, C2×C10, C2×C10 [×5], C3⋊C8 [×2], D12 [×2], D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C30, C30 [×2], C8⋊C22, C40 [×2], C2×C20, C2×C20, C5×D4, C5×D4 [×4], C5×Q8, C22×C10, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, C60 [×2], C60, S3×C10 [×4], C2×C30, C2×C30, C5×M4(2), C5×D8 [×2], C5×SD16 [×2], D4×C10, C5×C4○D4, D4⋊D6, C5×C3⋊C8 [×2], C5×D12 [×2], C5×D12, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, S3×C2×C10, C5×C8⋊C22, C5×C4.Dic3, C5×D4⋊S3 [×2], C5×Q82S3 [×2], C10×D12, C15×C4○D4, C5×D4⋊D6
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, D4⋊D6, C5×C3⋊D4 [×2], S3×C2×C10, C5×C8⋊C22, C10×C3⋊D4, C5×D4⋊D6

Smallest permutation representation of C5×D4⋊D6
On 120 points
Generators in S120
(1 49 37 25 14)(2 50 38 26 15)(3 51 39 27 13)(4 54 42 28 16)(5 52 40 29 17)(6 53 41 30 18)(7 57 45 31 19)(8 55 43 32 20)(9 56 44 33 21)(10 60 48 34 22)(11 58 46 35 23)(12 59 47 36 24)(61 113 101 89 78)(62 114 102 90 73)(63 109 97 85 74)(64 110 98 86 75)(65 111 99 87 76)(66 112 100 88 77)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(1 11 5 8)(2 12 6 9)(3 10 4 7)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 35 29 32)(26 36 30 33)(27 34 28 31)(37 46 40 43)(38 47 41 44)(39 48 42 45)(49 58 52 55)(50 59 53 56)(51 60 54 57)(61 71 64 68)(62 72 65 69)(63 67 66 70)(73 84 76 81)(74 79 77 82)(75 80 78 83)(85 91 88 94)(86 92 89 95)(87 93 90 96)(97 103 100 106)(98 104 101 107)(99 105 102 108)(109 115 112 118)(110 116 113 119)(111 117 114 120)
(1 68)(2 72)(3 70)(4 67)(5 71)(6 69)(7 63)(8 61)(9 65)(10 66)(11 64)(12 62)(13 82)(14 80)(15 84)(16 79)(17 83)(18 81)(19 74)(20 78)(21 76)(22 77)(23 75)(24 73)(25 92)(26 96)(27 94)(28 91)(29 95)(30 93)(31 85)(32 89)(33 87)(34 88)(35 86)(36 90)(37 104)(38 108)(39 106)(40 107)(41 105)(42 103)(43 101)(44 99)(45 97)(46 98)(47 102)(48 100)(49 116)(50 120)(51 118)(52 119)(53 117)(54 115)(55 113)(56 111)(57 109)(58 110)(59 114)(60 112)
(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 3)(4 5)(7 11)(8 10)(9 12)(13 14)(16 17)(19 23)(20 22)(21 24)(25 27)(28 29)(31 35)(32 34)(33 36)(37 39)(40 42)(43 48)(44 47)(45 46)(49 51)(52 54)(55 60)(56 59)(57 58)(61 70)(62 69)(63 68)(64 67)(65 72)(66 71)(73 81)(74 80)(75 79)(76 84)(77 83)(78 82)(85 92)(86 91)(87 96)(88 95)(89 94)(90 93)(97 104)(98 103)(99 108)(100 107)(101 106)(102 105)(109 116)(110 115)(111 120)(112 119)(113 118)(114 117)

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

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

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

105 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B5C5D6A6B6C6D8A8B10A10B10C10D10E10F10G10H10I10J10K10L10M···10T12A12B12C12D12E15A15B15C15D20A···20H20I20J20K20L30A30B30C30D30E···30P40A···40H60A···60H60I···60T
order1222223444555566668810101010101010101010101010···1012121212121515151520···20202020203030303030···3040···4060···6060···60
size11241212222411112444121211112222444412···122244422222···2444422224···412···122···24···4

105 irreducible representations

dim11111111111122222222222222224444
type++++++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D4D6D6D6C3⋊D4C3⋊D4C5×S3C5×D4C5×D4S3×C10S3×C10S3×C10C5×C3⋊D4C5×C3⋊D4C8⋊C22D4⋊D6C5×C8⋊C22C5×D4⋊D6
kernelC5×D4⋊D6C5×C4.Dic3C5×D4⋊S3C5×Q82S3C10×D12C15×C4○D4D4⋊D6C4.Dic3D4⋊S3Q82S3C2×D12C3×C4○D4C5×C4○D4C60C2×C30C2×C20C5×D4C5×Q8C20C2×C10C4○D4C12C2×C6C2×C4D4Q8C4C22C15C5C3C1
# reps11221144884411111122444444881248

Matrix representation of C5×D4⋊D6 in GL4(𝔽241) generated by

87000
08700
00870
00087
,
9919800
4314200
0014243
0019899
,
0014243
0019899
9919800
4314200
,
0100
24024000
000240
0011
,
24024000
0100
0014299
0019899
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[99,43,0,0,198,142,0,0,0,0,142,198,0,0,43,99],[0,0,99,43,0,0,198,142,142,198,0,0,43,99,0,0],[0,240,0,0,1,240,0,0,0,0,0,1,0,0,240,1],[240,0,0,0,240,1,0,0,0,0,142,198,0,0,99,99] >;

C5×D4⋊D6 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes D_6
% in TeX

G:=Group("C5xD4:D6");
// GroupNames label

G:=SmallGroup(480,828);
// by ID

G=gap.SmallGroup(480,828);
# by ID

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

G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽