Copied to
clipboard

## G = C3×D5×D8order 480 = 25·3·5

### Direct product of C3, D5 and D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D5×D8
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×D4×D5 — C3×D5×D8
 Lower central C5 — C10 — C20 — C3×D5×D8
 Upper central C1 — C6 — C12 — C3×D8

Generators and relations for C3×D5×D8
G = < a,b,c,d,e | a3=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 688 in 152 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C15, C2×C8, D8, D8, C2×D4, Dic5, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, C2×D8, C52C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C2×C24, C3×D8, C3×D8, C6×D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, C6×D8, C3×C52C8, C120, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, D5×C2×C6, D5×D8, D5×C24, C3×D40, C3×D4⋊D5, C15×D8, C3×D4×D5, C3×D5×D8
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, D8, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×D8, C22×D5, C3×D8, C6×D4, C6×D5, D4×D5, C6×D8, D5×C2×C6, D5×D8, C3×D4×D5, C3×D5×D8

Smallest permutation representation of C3×D5×D8
On 120 points
Generators in S120
(1 104 64)(2 97 57)(3 98 58)(4 99 59)(5 100 60)(6 101 61)(7 102 62)(8 103 63)(9 83 53)(10 84 54)(11 85 55)(12 86 56)(13 87 49)(14 88 50)(15 81 51)(16 82 52)(17 89 118)(18 90 119)(19 91 120)(20 92 113)(21 93 114)(22 94 115)(23 95 116)(24 96 117)(25 44 75)(26 45 76)(27 46 77)(28 47 78)(29 48 79)(30 41 80)(31 42 73)(32 43 74)(33 108 69)(34 109 70)(35 110 71)(36 111 72)(37 112 65)(38 105 66)(39 106 67)(40 107 68)
(1 105 42 20 54)(2 106 43 21 55)(3 107 44 22 56)(4 108 45 23 49)(5 109 46 24 50)(6 110 47 17 51)(7 111 48 18 52)(8 112 41 19 53)(9 103 65 80 91)(10 104 66 73 92)(11 97 67 74 93)(12 98 68 75 94)(13 99 69 76 95)(14 100 70 77 96)(15 101 71 78 89)(16 102 72 79 90)(25 115 86 58 40)(26 116 87 59 33)(27 117 88 60 34)(28 118 81 61 35)(29 119 82 62 36)(30 120 83 63 37)(31 113 84 64 38)(32 114 85 57 39)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 103)(10 104)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 110)(18 111)(19 112)(20 105)(21 106)(22 107)(23 108)(24 109)(33 116)(34 117)(35 118)(36 119)(37 120)(38 113)(39 114)(40 115)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 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 4)(2 3)(5 8)(6 7)(9 14)(10 13)(11 12)(15 16)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 38)(34 37)(35 36)(39 40)(41 46)(42 45)(43 44)(47 48)(49 54)(50 53)(51 52)(55 56)(57 58)(59 64)(60 63)(61 62)(65 70)(66 69)(67 68)(71 72)(73 76)(74 75)(77 80)(78 79)(81 82)(83 88)(84 87)(85 86)(89 90)(91 96)(92 95)(93 94)(97 98)(99 104)(100 103)(101 102)(105 108)(106 107)(109 112)(110 111)(113 116)(114 115)(117 120)(118 119)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 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 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 D8 D10 D10 C3×D4 C3×D4 C3×D5 C3×D8 C6×D5 C6×D5 D4×D5 D5×D8 C3×D4×D5 C3×D5×D8 kernel C3×D5×D8 D5×C24 C3×D40 C3×D4⋊D5 C15×D8 C3×D4×D5 D5×D8 C8×D5 D40 D4⋊D5 C5×D8 D4×D5 C3×Dic5 C6×D5 C3×D8 C3×D5 C24 C3×D4 Dic5 D10 D8 D5 C8 D4 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 2 4 2 4 2 2 4 8 4 8 2 4 4 8

Matrix representation of C3×D5×D8 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 0 1 0 0 240 51 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 230 11 0 0 230 230
,
 240 0 0 0 0 240 0 0 0 0 11 230 0 0 230 230
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,230,230,0,0,11,230],[240,0,0,0,0,240,0,0,0,0,11,230,0,0,230,230] >;

C3×D5×D8 in GAP, Magma, Sage, TeX

C_3\times D_5\times D_8
% in TeX

G:=Group("C3xD5xD8");
// GroupNames label

G:=SmallGroup(480,703);
// by ID

G=gap.SmallGroup(480,703);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,303,1271,648,102,18822]);
// Polycyclic

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

׿
×
𝔽