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## G = C3×C20.D4order 480 = 25·3·5

### Direct product of C3 and C20.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20.D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C3×C4.Dic5 — C3×C20.D4
 Lower central C5 — C10 — C2×C10 — C3×C20.D4
 Upper central C1 — C6 — C2×C12 — C6×D4

Generators and relations for C3×C20.D4
G = < a,b,c,d | a3=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b9, dcd-1=b15c3 >

Subgroups: 224 in 92 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, C2×C10, C2×C10, C24, C2×C12, C3×D4, C22×C6, C30, C30, C4.D4, C52C8, C2×C20, C5×D4, C22×C10, C3×M4(2), C6×D4, C60, C2×C30, C2×C30, C4.Dic5, D4×C10, C3×C4.D4, C3×C52C8, C2×C60, D4×C15, C22×C30, C20.D4, C3×C4.Dic5, D4×C30, C3×C20.D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, Dic5, D10, C2×C12, C3×D4, C3×D5, C4.D4, C2×Dic5, C5⋊D4, C3×C22⋊C4, C3×Dic5, C6×D5, C23.D5, C3×C4.D4, C6×Dic5, C3×C5⋊D4, C20.D4, C3×C23.D5, C3×C20.D4

Smallest permutation representation of C3×C20.D4
On 120 points
Generators in S120
(1 56 29)(2 57 30)(3 58 31)(4 59 32)(5 60 33)(6 41 34)(7 42 35)(8 43 36)(9 44 37)(10 45 38)(11 46 39)(12 47 40)(13 48 21)(14 49 22)(15 50 23)(16 51 24)(17 52 25)(18 53 26)(19 54 27)(20 55 28)(61 106 96)(62 107 97)(63 108 98)(64 109 99)(65 110 100)(66 111 81)(67 112 82)(68 113 83)(69 114 84)(70 115 85)(71 116 86)(72 117 87)(73 118 88)(74 119 89)(75 120 90)(76 101 91)(77 102 92)(78 103 93)(79 104 94)(80 105 95)
(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 61 6 76 11 71 16 66)(2 80 7 75 12 70 17 65)(3 79 8 74 13 69 18 64)(4 78 9 73 14 68 19 63)(5 77 10 72 15 67 20 62)(21 84 26 99 31 94 36 89)(22 83 27 98 32 93 37 88)(23 82 28 97 33 92 38 87)(24 81 29 96 34 91 39 86)(25 100 30 95 35 90 40 85)(41 101 46 116 51 111 56 106)(42 120 47 115 52 110 57 105)(43 119 48 114 53 109 58 104)(44 118 49 113 54 108 59 103)(45 117 50 112 55 107 60 102)
(1 71 6 76 11 61 16 66)(2 80 7 65 12 70 17 75)(3 69 8 74 13 79 18 64)(4 78 9 63 14 68 19 73)(5 67 10 72 15 77 20 62)(21 94 26 99 31 84 36 89)(22 83 27 88 32 93 37 98)(23 92 28 97 33 82 38 87)(24 81 29 86 34 91 39 96)(25 90 30 95 35 100 40 85)(41 101 46 106 51 111 56 116)(42 110 47 115 52 120 57 105)(43 119 48 104 53 109 58 114)(44 108 49 113 54 118 59 103)(45 117 50 102 55 107 60 112)

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

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

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

93 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 15A 15B 15C 15D 20A 20B 20C 20D 24A ··· 24H 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 3 3 4 4 5 5 6 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 20 20 20 24 ··· 24 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 4 4 1 1 2 2 2 2 1 1 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 ··· 4 2 2 2 2 2 2 2 2 4 4 4 4 20 ··· 20 2 ··· 2 4 ··· 4 4 ··· 4

93 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 D4 D5 D10 Dic5 C3×D4 C3×D5 C5⋊D4 C6×D5 C3×Dic5 C3×C5⋊D4 C4.D4 C3×C4.D4 C20.D4 C3×C20.D4 kernel C3×C20.D4 C3×C4.Dic5 D4×C30 C20.D4 C22×C30 C4.Dic5 D4×C10 C22×C10 C60 C6×D4 C2×C12 C22×C6 C20 C2×D4 C12 C2×C4 C23 C4 C15 C5 C3 C1 # reps 1 2 1 2 4 4 2 8 2 2 2 4 4 4 8 4 8 16 1 2 4 8

Matrix representation of C3×C20.D4 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 0 98 0 0 143 0 0 0 0 0 0 91 0 0 150 0
,
 0 0 1 0 0 0 0 240 0 1 0 0 1 0 0 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 240 0 0 0
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,143,0,0,98,0,0,0,0,0,0,150,0,0,91,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;

C3×C20.D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}.D_4
% in TeX

G:=Group("C3xC20.D4");
// GroupNames label

G:=SmallGroup(480,111);
// by ID

G=gap.SmallGroup(480,111);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,850,136,2524,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

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