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G = C6×D20order 240 = 24·3·5

Direct product of C6 and D20

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

Aliases: C6×D20, C305D4, C129D10, C6010C22, C30.40C23, C51(C6×D4), C42(C6×D5), (C2×C20)⋊3C6, C202(C2×C6), (C2×C60)⋊8C2, (C2×C12)⋊6D5, C101(C3×D4), C1511(C2×D4), D101(C2×C6), (C2×C6).38D10, (C6×D5)⋊9C22, (C22×D5)⋊2C6, C10.3(C22×C6), C6.40(C22×D5), C22.10(C6×D5), (C2×C30).39C22, (D5×C2×C6)⋊5C2, C2.4(D5×C2×C6), (C2×C4)⋊2(C3×D5), (C2×C10).10(C2×C6), SmallGroup(240,157)

Series: Derived Chief Lower central Upper central

C1C10 — C6×D20
C1C5C10C30C6×D5D5×C2×C6 — C6×D20
C5C10 — C6×D20
C1C2×C6C2×C12

Generators and relations for C6×D20
 G = < a,b,c | a6=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 356 in 108 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, C6, C6 [×2], C6 [×4], C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], C12 [×2], C2×C6, C2×C6 [×8], C15, C2×D4, C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C12, C3×D4 [×4], C22×C6 [×2], C3×D5 [×4], C30, C30 [×2], D20 [×4], C2×C20, C22×D5 [×2], C6×D4, C60 [×2], C6×D5 [×4], C6×D5 [×4], C2×C30, C2×D20, C3×D20 [×4], C2×C60, D5×C2×C6 [×2], C6×D20
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, D20 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×D20, C3×D20 [×2], D5×C2×C6, C6×D20

Smallest permutation representation of C6×D20
On 120 points
Generators in S120
(1 29 61 50 92 109)(2 30 62 51 93 110)(3 31 63 52 94 111)(4 32 64 53 95 112)(5 33 65 54 96 113)(6 34 66 55 97 114)(7 35 67 56 98 115)(8 36 68 57 99 116)(9 37 69 58 100 117)(10 38 70 59 81 118)(11 39 71 60 82 119)(12 40 72 41 83 120)(13 21 73 42 84 101)(14 22 74 43 85 102)(15 23 75 44 86 103)(16 24 76 45 87 104)(17 25 77 46 88 105)(18 26 78 47 89 106)(19 27 79 48 90 107)(20 28 80 49 91 108)
(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 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 60)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 82)(29 81)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 112)(68 111)(69 110)(70 109)(71 108)(72 107)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 120)(80 119)

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

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

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

C6×D20 is a maximal subgroup of
C60.28D4  C30.D8  C6.D40  (C2×C60)⋊C4  C60.36D4  D6030C22  C60.44D4  C60.88D4  (C6×D5).D4  Dic15⋊D4  Dic3⋊D20  D208Dic3  C604D4  C12⋊D20  C6010D4  C122D20  D64D20  D2025D6  C6×D4×D5

78 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B5A5B6A···6F6G···6N10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122222223344556···66···610···10121212121515151520···2030···3060···60
size1111101010101122221···110···102···2222222222···22···22···2

78 irreducible representations

dim111111112222222222
type+++++++++
imageC1C2C2C2C3C6C6C6D4D5D10D10C3×D4C3×D5D20C6×D5C6×D5C3×D20
kernelC6×D20C3×D20C2×C60D5×C2×C6C2×D20D20C2×C20C22×D5C30C2×C12C12C2×C6C10C2×C4C6C4C22C2
# reps1412282422424488416

Matrix representation of C6×D20 in GL3(𝔽61) generated by

1400
0140
0014
,
6000
05929
03254
,
6000
05432
0297
G:=sub<GL(3,GF(61))| [14,0,0,0,14,0,0,0,14],[60,0,0,0,59,32,0,29,54],[60,0,0,0,54,29,0,32,7] >;

C6×D20 in GAP, Magma, Sage, TeX

C_6\times D_{20}
% in TeX

G:=Group("C6xD20");
// GroupNames label

G:=SmallGroup(240,157);
// by ID

G=gap.SmallGroup(240,157);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,506,122,6917]);
// Polycyclic

G:=Group<a,b,c|a^6=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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