Copied to
clipboard

## G = C5×C12.46D4order 480 = 25·3·5

### Direct product of C5 and C12.46D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C12.46D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C10×D12 — C5×C12.46D4
 Lower central C3 — C6 — C2×C6 — C5×C12.46D4
 Upper central C1 — C10 — C2×C20 — C5×M4(2)

Generators and relations for C5×C12.46D4
G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >

Subgroups: 292 in 92 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], D6 [×4], C2×C6, C15, M4(2), M4(2), C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], C5×S3 [×2], C30, C30, C4.D4, C40 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], S3×C10 [×4], C2×C30, C5×M4(2), C5×M4(2), D4×C10, C12.46D4, C5×C3⋊C8, C120, C5×D12 [×2], C2×C60, S3×C2×C10 [×2], C5×C4.D4, C5×C4.Dic3, C15×M4(2), C10×D12, C5×C12.46D4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4.D4, C2×C20, C5×D4 [×2], D6⋊C4, S3×C10, C5×C22⋊C4, C12.46D4, S3×C20, C5×D12, C5×C3⋊D4, C5×C4.D4, C5×D6⋊C4, C5×C12.46D4

Smallest permutation representation of C5×C12.46D4
On 120 points
Generators in S120
(1 58 41 51 24)(2 59 42 52 17)(3 60 43 53 18)(4 61 44 54 19)(5 62 45 55 20)(6 63 46 56 21)(7 64 47 49 22)(8 57 48 50 23)(9 85 110 75 36)(10 86 111 76 37)(11 87 112 77 38)(12 88 105 78 39)(13 81 106 79 40)(14 82 107 80 33)(15 83 108 73 34)(16 84 109 74 35)(25 101 72 92 117)(26 102 65 93 118)(27 103 66 94 119)(28 104 67 95 120)(29 97 68 96 113)(30 98 69 89 114)(31 99 70 90 115)(32 100 71 91 116)
(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 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(98 102)(100 104)(105 109)(107 111)(114 118)(116 120)
(1 99 75)(2 100 76)(3 101 77)(4 102 78)(5 103 79)(6 104 80)(7 97 73)(8 98 74)(9 41 90)(10 42 91)(11 43 92)(12 44 93)(13 45 94)(14 46 95)(15 47 96)(16 48 89)(17 32 111)(18 25 112)(19 26 105)(20 27 106)(21 28 107)(22 29 108)(23 30 109)(24 31 110)(33 63 67)(34 64 68)(35 57 69)(36 58 70)(37 59 71)(38 60 72)(39 61 65)(40 62 66)(49 113 83)(50 114 84)(51 115 85)(52 116 86)(53 117 87)(54 118 88)(55 119 81)(56 120 82)
(2 6)(3 7)(9 90)(10 95)(11 96)(12 93)(13 94)(14 91)(15 92)(16 89)(17 21)(18 22)(25 108)(26 105)(27 106)(28 111)(29 112)(30 109)(31 110)(32 107)(33 71)(34 72)(35 69)(36 70)(37 67)(38 68)(39 65)(40 66)(42 46)(43 47)(49 53)(52 56)(59 63)(60 64)(73 101)(74 98)(75 99)(76 104)(77 97)(78 102)(79 103)(80 100)(81 119)(82 116)(83 117)(84 114)(85 115)(86 120)(87 113)(88 118)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 5A 5B 5C 5D 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 12A 12B 12C 15A 15B 15C 15D 20A ··· 20H 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 40I ··· 40P 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 2 3 4 4 5 5 5 5 6 6 8 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 12 12 12 15 15 15 15 20 ··· 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 12 12 2 2 2 1 1 1 1 2 4 4 4 12 12 1 1 1 1 2 2 2 2 12 ··· 12 2 2 4 2 2 2 2 2 ··· 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 12 ··· 12 2 ··· 2 4 4 4 4 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 D4 D6 D12 C3⋊D4 C4×S3 C5×S3 C5×D4 S3×C10 C5×D12 C5×C3⋊D4 S3×C20 C4.D4 C12.46D4 C5×C4.D4 C5×C12.46D4 kernel C5×C12.46D4 C5×C4.Dic3 C15×M4(2) C10×D12 S3×C2×C10 C12.46D4 C4.Dic3 C3×M4(2) C2×D12 C22×S3 C5×M4(2) C60 C2×C20 C20 C20 C2×C10 M4(2) C12 C2×C4 C4 C4 C22 C15 C5 C3 C1 # reps 1 1 1 1 4 4 4 4 4 16 1 2 1 2 2 2 4 8 4 8 8 8 1 2 4 8

Matrix representation of C5×C12.46D4 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 98 0 0 0 0 98
,
 90 81 226 153 205 106 120 34 198 86 47 16 155 43 29 239
,
 1 0 125 189 0 1 72 8 0 0 240 0 0 0 0 240
,
 240 1 10 88 240 0 98 231 0 0 240 1 0 0 240 0
,
 1 240 205 127 0 240 169 36 0 0 1 240 0 0 0 240
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[90,205,198,155,81,106,86,43,226,120,47,29,153,34,16,239],[1,0,0,0,0,1,0,0,125,72,240,0,189,8,0,240],[240,240,0,0,1,0,0,0,10,98,240,240,88,231,1,0],[1,0,0,0,240,240,0,0,205,169,1,0,127,36,240,240] >;

C5×C12.46D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}._{46}D_4
% in TeX

G:=Group("C5xC12.46D4");
// GroupNames label

G:=SmallGroup(480,142);
// by ID

G=gap.SmallGroup(480,142);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,2803,136,2111,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽