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## G = (C2×C60)⋊C4order 480 = 25·3·5

### 2nd semidirect product of C2×C60 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — (C2×C60)⋊C4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C2×C6 — D10.D6 — (C2×C60)⋊C4
 Lower central C15 — C30 — C2×C30 — (C2×C60)⋊C4
 Upper central C1 — C2 — C22 — C2×C4

Generators and relations for (C2×C60)⋊C4
G = < a,b,c | a2=b60=c4=1, ab=ba, cac-1=ab30, cbc-1=ab47 >

Subgroups: 716 in 104 conjugacy classes, 29 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C23⋊C4, D20, C2×C20, C2×F5, C22×D5, C6.D4, C6×D4, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C22⋊F5, C2×D20, C23.7D6, C3×D20, C2×C60, C2×C3⋊F5, D5×C2×C6, D10.D4, D10.D6, C6×D20, (C2×C60)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C23⋊C4, C2×F5, C6.D4, C3⋊F5, C22⋊F5, C23.7D6, C2×C3⋊F5, D10.D4, D10.D6, (C2×C60)⋊C4

Smallest permutation representation of (C2×C60)⋊C4
On 120 points
Generators in S120
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(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)
(2 114 20 108)(3 47 39 35)(4 100 58 82)(5 33 17 9)(6 86 36 116)(7 19 55 43)(8 72 14 90)(10 118 52 64)(11 51)(12 104 30 98)(13 37 49 25)(15 23 27 59)(16 76 46 106)(18 62 24 80)(21 41)(22 94 40 88)(26 66 56 96)(28 112 34 70)(29 45 53 57)(32 84 50 78)(38 102 44 120)(42 74 60 68)(48 92 54 110)(61 91)(63 77 99 65)(67 109 115 73)(69 95 93 107)(71 81)(75 113 87 89)(79 85 103 97)(83 117 119 105)(101 111)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - - + + + + + + image C1 C2 C2 C4 C4 S3 D4 Dic3 Dic3 D6 C3⋊D4 F5 C23⋊C4 C2×F5 C3⋊F5 C22⋊F5 C23.7D6 C2×C3⋊F5 D10.D4 D10.D6 (C2×C60)⋊C4 kernel (C2×C60)⋊C4 D10.D6 C6×D20 C2×C60 D5×C2×C6 C2×D20 C6×D5 C2×C20 C22×D5 C22×D5 D10 C2×C12 C15 C2×C6 C2×C4 C6 C5 C22 C3 C2 C1 # reps 1 2 1 2 2 1 2 1 1 1 4 1 1 1 2 2 2 2 4 4 8

Matrix representation of (C2×C60)⋊C4 in GL8(𝔽61)

 52 43 0 0 0 0 0 0 18 9 0 0 0 0 0 0 0 0 52 43 0 0 0 0 0 0 18 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 60 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 6 6 0 34 0 0 0 0 27 33 33 27 0 0 0 0 34 0 6 6 0 0 0 0 55 28 55 0
,
 1 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 9 18 0 0 0 0 0 0 9 52 0 0 0 0 0 0 0 0 6 34 6 0 0 0 0 0 55 55 0 28 0 0 0 0 28 0 55 55 0 0 0 0 0 6 34 6

G:=sub<GL(8,GF(61))| [52,18,0,0,0,0,0,0,43,9,0,0,0,0,0,0,0,0,52,18,0,0,0,0,0,0,43,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,0,6,27,34,55,0,0,0,0,6,33,0,28,0,0,0,0,0,33,6,55,0,0,0,0,34,27,6,0],[1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,9,9,0,0,0,0,0,0,18,52,0,0,0,0,0,0,0,0,6,55,28,0,0,0,0,0,34,55,0,6,0,0,0,0,6,0,55,34,0,0,0,0,0,28,55,6] >;

(C2×C60)⋊C4 in GAP, Magma, Sage, TeX

(C_2\times C_{60})\rtimes C_4
% in TeX

G:=Group("(C2xC60):C4");
// GroupNames label

G:=SmallGroup(480,304);
// by ID

G=gap.SmallGroup(480,304);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,675,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c|a^2=b^60=c^4=1,a*b=b*a,c*a*c^-1=a*b^30,c*b*c^-1=a*b^47>;
// generators/relations

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