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G = C6xD20order 240 = 24·3·5

Direct product of C6 and D20

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

Aliases: C6xD20, C30:5D4, C12:9D10, C60:10C22, C30.40C23, C5:1(C6xD4), C4:2(C6xD5), (C2xC20):3C6, C20:2(C2xC6), (C2xC60):8C2, (C2xC12):6D5, C10:1(C3xD4), C15:11(C2xD4), D10:1(C2xC6), (C2xC6).38D10, (C6xD5):9C22, (C22xD5):2C6, C10.3(C22xC6), C6.40(C22xD5), C22.10(C6xD5), (C2xC30).39C22, (D5xC2xC6):5C2, C2.4(D5xC2xC6), (C2xC4):2(C3xD5), (C2xC10).10(C2xC6), SmallGroup(240,157)

Series: Derived Chief Lower central Upper central

C1C10 — C6xD20
C1C5C10C30C6xD5D5xC2xC6 — C6xD20
C5C10 — C6xD20
C1C2xC6C2xC12

Generators and relations for C6xD20
 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, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2xC4, D4, C23, D5, C10, C10, C12, C2xC6, C2xC6, C15, C2xD4, C20, D10, D10, C2xC10, C2xC12, C3xD4, C22xC6, C3xD5, C30, C30, D20, C2xC20, C22xD5, C6xD4, C60, C6xD5, C6xD5, C2xC30, C2xD20, C3xD20, C2xC60, D5xC2xC6, C6xD20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2xC6, C2xD4, D10, C3xD4, C22xC6, C3xD5, D20, C22xD5, C6xD4, C6xD5, C2xD20, C3xD20, D5xC2xC6, C6xD20

Smallest permutation representation of C6xD20
On 120 points
Generators in S120
(1 104 70 83 52 31)(2 105 71 84 53 32)(3 106 72 85 54 33)(4 107 73 86 55 34)(5 108 74 87 56 35)(6 109 75 88 57 36)(7 110 76 89 58 37)(8 111 77 90 59 38)(9 112 78 91 60 39)(10 113 79 92 41 40)(11 114 80 93 42 21)(12 115 61 94 43 22)(13 116 62 95 44 23)(14 117 63 96 45 24)(15 118 64 97 46 25)(16 119 65 98 47 26)(17 120 66 99 48 27)(18 101 67 100 49 28)(19 102 68 81 50 29)(20 103 69 82 51 30)
(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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 104)(42 103)(43 102)(44 101)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)

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

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

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

C6xD20 is a maximal subgroup of
C60.28D4  C30.D8  C6.D40  (C2xC60):C4  C60.36D4  D60:30C22  C60.44D4  C60.88D4  (C6xD5).D4  Dic15:D4  Dic3:D20  D20:8Dic3  C60:4D4  C12:D20  C60:10D4  C12:2D20  D6:4D20  D20:25D6  C6xD4xD5

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+++++++++
imageC1C2C2C2C3C6C6C6D4D5D10D10C3xD4C3xD5D20C6xD5C6xD5C3xD20
kernelC6xD20C3xD20C2xC60D5xC2xC6C2xD20D20C2xC20C22xD5C30C2xC12C12C2xC6C10C2xC4C6C4C22C2
# reps1412282422424488416

Matrix representation of C6xD20 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] >;

C6xD20 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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