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## G = (C2×C12)⋊6F5order 480 = 25·3·5

### 4th semidirect product of C2×C12 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — (C2×C12)⋊6F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C2×C3⋊F5 — C4×C3⋊F5 — (C2×C12)⋊6F5
 Lower central C15 — C30 — (C2×C12)⋊6F5
 Upper central C1 — C4 — C2×C4

Generators and relations for (C2×C12)⋊6F5
G = < a,b,c,d | a2=b12=c5=d4=1, ab=ba, ac=ca, dad-1=ab6, bc=cb, dbd-1=b5, dcd-1=c3 >

Subgroups: 716 in 152 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C23.26D6, D5×C12, C6×Dic5, C2×C60, C2×C3⋊F5, D5×C2×C6, D10.C23, C4×C3⋊F5, C60⋊C4, D10.D6, D5×C2×C12, (C2×C12)⋊6F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, F5, C2×Dic3, C22×S3, C42⋊C2, C2×F5, C4○D12, C22×Dic3, C3⋊F5, C22×F5, C23.26D6, C2×C3⋊F5, D10.C23, C22×C3⋊F5, (C2×C12)⋊6F5

Smallest permutation representation of (C2×C12)⋊6F5
On 120 points
Generators in S120
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 109)(13 106)(14 107)(15 108)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)
(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 71 90 84 100)(2 72 91 73 101)(3 61 92 74 102)(4 62 93 75 103)(5 63 94 76 104)(6 64 95 77 105)(7 65 96 78 106)(8 66 85 79 107)(9 67 86 80 108)(10 68 87 81 97)(11 69 88 82 98)(12 70 89 83 99)(13 116 36 58 48)(14 117 25 59 37)(15 118 26 60 38)(16 119 27 49 39)(17 120 28 50 40)(18 109 29 51 41)(19 110 30 52 42)(20 111 31 53 43)(21 112 32 54 44)(22 113 33 55 45)(23 114 34 56 46)(24 115 35 57 47)
(2 6)(3 11)(5 9)(8 12)(13 42 36 52)(14 47 25 57)(15 40 26 50)(16 45 27 55)(17 38 28 60)(18 43 29 53)(19 48 30 58)(20 41 31 51)(21 46 32 56)(22 39 33 49)(23 44 34 54)(24 37 35 59)(61 88 102 82)(62 93 103 75)(63 86 104 80)(64 91 105 73)(65 96 106 78)(66 89 107 83)(67 94 108 76)(68 87 97 81)(69 92 98 74)(70 85 99 79)(71 90 100 84)(72 95 101 77)(109 111)(110 116)(112 114)(113 119)(115 117)(118 120)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + - + - - + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 Dic3 D6 Dic3 Dic3 D6 C4○D4 C4○D12 F5 C2×F5 C2×F5 C3⋊F5 C2×C3⋊F5 C2×C3⋊F5 D10.C23 (C2×C12)⋊6F5 kernel (C2×C12)⋊6F5 C4×C3⋊F5 C60⋊C4 D10.D6 D5×C2×C12 D5×C12 C6×Dic5 C2×C60 C2×C4×D5 C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 C3×D5 D5 C2×C12 C12 C2×C6 C2×C4 C4 C22 C3 C1 # reps 1 2 2 2 1 4 2 2 1 2 2 1 1 1 4 8 1 2 1 2 4 2 4 8

Matrix representation of (C2×C12)⋊6F5 in GL4(𝔽61) generated by

 7 0 14 14 47 54 47 0 0 47 54 47 14 14 0 7
,
 8 0 5 5 56 3 56 0 0 56 3 56 5 5 0 8
,
 60 60 60 60 1 0 0 0 0 1 0 0 0 0 1 0
,
 60 0 0 0 0 0 0 60 0 60 0 0 1 1 1 1
G:=sub<GL(4,GF(61))| [7,47,0,14,0,54,47,14,14,47,54,0,14,0,47,7],[8,56,0,5,0,3,56,5,5,56,3,0,5,0,56,8],[60,1,0,0,60,0,1,0,60,0,0,1,60,0,0,0],[60,0,0,1,0,0,60,1,0,0,0,1,0,60,0,1] >;

(C2×C12)⋊6F5 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes_6F_5
% in TeX

G:=Group("(C2xC12):6F5");
// GroupNames label

G:=SmallGroup(480,1065);
// by ID

G=gap.SmallGroup(480,1065);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,2693,14118,2379]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^6,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations

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