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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 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C24 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×3], C4.D4, C52C8 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C3×M4(2) [×2], C6×D4, C60 [×2], C2×C30, C2×C30 [×4], C4.Dic5 [×2], D4×C10, C3×C4.D4, C3×C52C8 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C20.D4, C3×C4.Dic5 [×2], D4×C30, C3×C20.D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, Dic5 [×2], D10, C2×C12, C3×D4 [×2], C3×D5, C4.D4, C2×Dic5, C5⋊D4 [×2], C3×C22⋊C4, C3×Dic5 [×2], C6×D5, C23.D5, C3×C4.D4, C6×Dic5, C3×C5⋊D4 [×2], C20.D4, C3×C23.D5, C3×C20.D4

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