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## G = C6×D20order 240 = 24·3·5

### Direct product of C6 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×D20
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — C6×D20
 Lower central C5 — C10 — C6×D20
 Upper central C1 — C2×C6 — C2×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, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C2×D4, C20, D10, D10, C2×C10, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, D20, C2×C20, C22×D5, C6×D4, C60, C6×D5, C6×D5, C2×C30, C2×D20, C3×D20, C2×C60, D5×C2×C6, C6×D20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, D20, C22×D5, C6×D4, C6×D5, C2×D20, C3×D20, D5×C2×C6, C6×D20

Smallest permutation representation of C6×D20
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)]])

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 6A ··· 6F 6G ··· 6N 10A ··· 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 60A ··· 60P order 1 2 2 2 2 2 2 2 3 3 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 10 10 1 1 2 2 2 2 1 ··· 1 10 ··· 10 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 D10 D10 C3×D4 C3×D5 D20 C6×D5 C6×D5 C3×D20 kernel C6×D20 C3×D20 C2×C60 D5×C2×C6 C2×D20 D20 C2×C20 C22×D5 C30 C2×C12 C12 C2×C6 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 2 2 4 2 4 4 8 8 4 16

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

 14 0 0 0 14 0 0 0 14
,
 60 0 0 0 59 29 0 32 54
,
 60 0 0 0 54 32 0 29 7
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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