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G = C3×D207C4order 480 = 25·3·5

Direct product of C3 and D207C4

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

Aliases: C3×D207C4, D207C12, C60.232D4, Dic107C12, C1519C4≀C2, C4.3(D5×C12), (C3×D20)⋊16C4, (C4×Dic5)⋊1C6, C4○D20.2C6, (C2×C30).78D4, C20.54(C3×D4), (C2×C6).25D20, C12.46(C4×D5), C60.161(C2×C4), C20.27(C2×C12), (C3×M4(2))⋊8D5, (C5×M4(2))⋊8C6, M4(2)⋊4(C3×D5), C22.3(C3×D20), (C12×Dic5)⋊13C2, (C3×Dic10)⋊16C4, (C2×C12).353D10, C12.122(C5⋊D4), C30.89(C22⋊C4), (C15×M4(2))⋊16C2, (C2×C60).278C22, C6.42(D10⋊C4), C54(C3×C4≀C2), (C2×C10).1(C3×D4), (C2×C4).32(C6×D5), C4.29(C3×C5⋊D4), (C2×C20).14(C2×C6), (C3×C4○D20).8C2, C10.21(C3×C22⋊C4), C2.11(C3×D10⋊C4), SmallGroup(480,103)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D207C4
C1C5C10C20C2×C20C2×C60C3×C4○D20 — C3×D207C4
C5C10C20 — C3×D207C4
C1C12C2×C12C3×M4(2)

Generators and relations for C3×D207C4
 G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b9, dcd-1=b3c >

Subgroups: 320 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, D5, C10, C10, C12 [×2], C12 [×3], C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C24, C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8, C3×D5, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5 [×3], C60 [×2], C6×D5, C2×C30, C4×Dic5, C5×M4(2), C4○D20, C3×C4≀C2, C120, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D207C4, C12×Dic5, C15×M4(2), C3×C4○D20, C3×D207C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D10, C2×C12, C3×D4 [×2], C3×D5, C4≀C2, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4≀C2, D5×C12, C3×D20, C3×C5⋊D4, D207C4, C3×D10⋊C4, C3×D207C4

Smallest permutation representation of C3×D207C4
On 120 points
Generators in S120
(1 46 31)(2 47 32)(3 48 33)(4 49 34)(5 50 35)(6 51 36)(7 52 37)(8 53 38)(9 54 39)(10 55 40)(11 56 21)(12 57 22)(13 58 23)(14 59 24)(15 60 25)(16 41 26)(17 42 27)(18 43 28)(19 44 29)(20 45 30)(61 111 86)(62 112 87)(63 113 88)(64 114 89)(65 115 90)(66 116 91)(67 117 92)(68 118 93)(69 119 94)(70 120 95)(71 101 96)(72 102 97)(73 103 98)(74 104 99)(75 105 100)(76 106 81)(77 107 82)(78 108 83)(79 109 84)(80 110 85)
(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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 95)(22 94)(23 93)(24 92)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 81)(36 100)(37 99)(38 98)(39 97)(40 96)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 120)(57 119)(58 118)(59 117)(60 116)
(1 16 11 6)(2 5 12 15)(3 14 13 4)(7 10 17 20)(8 19 18 9)(21 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(62 70)(63 79)(64 68)(65 77)(67 75)(69 73)(72 80)(74 78)(82 90)(83 99)(84 88)(85 97)(87 95)(89 93)(92 100)(94 98)(102 110)(103 119)(104 108)(105 117)(107 115)(109 113)(112 120)(114 118)

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

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

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

102 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F8A8B10A10B10C10D12A12B12C12D12E12F12G···12N12O12P15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1222334444444455666666881010101012121212121212···1212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size11220111121010101020221122202044224411112210···10202022222222444444222244444···42···244444···4

102 irreducible representations

dim111111111111222222222222222244
type+++++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12D4D4D5D10C3×D4C3×D4C3×D5C4≀C2C4×D5C5⋊D4D20C6×D5C3×C4≀C2D5×C12C3×C5⋊D4C3×D20D207C4C3×D207C4
kernelC3×D207C4C12×Dic5C15×M4(2)C3×C4○D20D207C4C3×Dic10C3×D20C4×Dic5C5×M4(2)C4○D20Dic10D20C60C2×C30C3×M4(2)C2×C12C20C2×C10M4(2)C15C12C12C2×C6C2×C4C5C4C4C22C3C1
# reps111122222244112222444444888848

Matrix representation of C3×D207C4 in GL4(𝔽241) generated by

225000
022500
002250
000225
,
24024000
19119000
006447
000177
,
05200
51000
002400
00541
,
515200
19119000
00177155
0001
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[240,191,0,0,240,190,0,0,0,0,64,0,0,0,47,177],[0,51,0,0,52,0,0,0,0,0,240,54,0,0,0,1],[51,191,0,0,52,190,0,0,0,0,177,0,0,0,155,1] >;

C3×D207C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_7C_4
% in TeX

G:=Group("C3xD20:7C4");
// GroupNames label

G:=SmallGroup(480,103);
// by ID

G=gap.SmallGroup(480,103);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,136,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^3*c>;
// generators/relations

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