Copied to
clipboard

## G = C5×S3×M4(2)  order 480 = 25·3·5

### Direct product of C5, S3 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×S3×M4(2)
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — S3×C2×C20 — C5×S3×M4(2)
 Lower central C3 — C6 — C5×S3×M4(2)
 Upper central C1 — C20 — C5×M4(2)

Generators and relations for C5×S3×M4(2)
G = < a,b,c,d,e | a5=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 260 in 136 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], S3, C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C10, C10 [×4], Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C5×S3, C30, C30, C2×M4(2), C40 [×2], C40 [×2], C2×C20, C2×C20 [×5], C22×C10, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C40 [×2], C5×M4(2), C5×M4(2) [×3], C22×C20, S3×M4(2), C5×C3⋊C8 [×2], C120 [×2], S3×C20 [×4], C10×Dic3, C2×C60, S3×C2×C10, C10×M4(2), S3×C40 [×2], C5×C8⋊S3 [×2], C5×C4.Dic3, C15×M4(2), S3×C2×C20, C5×S3×M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], M4(2) [×2], C22×C4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C2×M4(2), C2×C20 [×6], C22×C10, S3×C2×C4, S3×C10 [×3], C5×M4(2) [×2], C22×C20, S3×M4(2), S3×C20 [×2], S3×C2×C10, C10×M4(2), S3×C2×C20, C5×S3×M4(2)

Smallest permutation representation of C5×S3×M4(2)
On 120 points
Generators in S120
(1 15 35 108 29)(2 16 36 109 30)(3 9 37 110 31)(4 10 38 111 32)(5 11 39 112 25)(6 12 40 105 26)(7 13 33 106 27)(8 14 34 107 28)(17 43 58 82 51)(18 44 59 83 52)(19 45 60 84 53)(20 46 61 85 54)(21 47 62 86 55)(22 48 63 87 56)(23 41 64 88 49)(24 42 57 81 50)(65 93 118 77 102)(66 94 119 78 103)(67 95 120 79 104)(68 96 113 80 97)(69 89 114 73 98)(70 90 115 74 99)(71 91 116 75 100)(72 92 117 76 101)
(1 19 99)(2 20 100)(3 21 101)(4 22 102)(5 23 103)(6 24 104)(7 17 97)(8 18 98)(9 47 72)(10 48 65)(11 41 66)(12 42 67)(13 43 68)(14 44 69)(15 45 70)(16 46 71)(25 49 78)(26 50 79)(27 51 80)(28 52 73)(29 53 74)(30 54 75)(31 55 76)(32 56 77)(33 58 96)(34 59 89)(35 60 90)(36 61 91)(37 62 92)(38 63 93)(39 64 94)(40 57 95)(81 120 105)(82 113 106)(83 114 107)(84 115 108)(85 116 109)(86 117 110)(87 118 111)(88 119 112)
(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(49 78)(50 79)(51 80)(52 73)(53 74)(54 75)(55 76)(56 77)(57 95)(58 96)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(81 120)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)
(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)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)(81 85)(83 87)(89 93)(91 95)(98 102)(100 104)(105 109)(107 111)(114 118)(116 120)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 10Q 10R 10S 10T 12A 12B 12C 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M ··· 20T 20U 20V 20W 20X 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40P 40Q ··· 40AF 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 10 10 10 10 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 ··· 20 20 20 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 3 3 6 2 1 1 2 3 3 6 1 1 1 1 2 4 2 2 2 2 6 6 6 6 1 1 1 1 2 2 2 2 3 ··· 3 6 6 6 6 2 2 4 2 2 2 2 1 ··· 1 2 2 2 2 3 ··· 3 6 6 6 6 4 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 6 ··· 6 2 ··· 2 4 4 4 4 4 ··· 4

150 irreducible representations

 dim 1 1 1 1 1 1 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 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C10 C20 C20 C20 S3 D6 D6 M4(2) C4×S3 C4×S3 C5×S3 S3×C10 S3×C10 C5×M4(2) S3×C20 S3×C20 S3×M4(2) C5×S3×M4(2) kernel C5×S3×M4(2) S3×C40 C5×C8⋊S3 C5×C4.Dic3 C15×M4(2) S3×C2×C20 S3×C20 C10×Dic3 S3×C2×C10 S3×M4(2) S3×C8 C8⋊S3 C4.Dic3 C3×M4(2) S3×C2×C4 C4×S3 C2×Dic3 C22×S3 C5×M4(2) C40 C2×C20 C5×S3 C20 C2×C10 M4(2) C8 C2×C4 S3 C4 C22 C5 C1 # reps 1 2 2 1 1 1 4 2 2 4 8 8 4 4 4 16 8 8 1 2 1 4 2 2 4 8 4 16 8 8 2 8

Matrix representation of C5×S3×M4(2) in GL4(𝔽241) generated by

 205 0 0 0 0 205 0 0 0 0 91 0 0 0 0 91
,
 240 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 46 239 0 0 126 195
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 46 240
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,91,0,0,0,0,91],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,46,126,0,0,239,195],[1,0,0,0,0,1,0,0,0,0,1,46,0,0,0,240] >;

C5×S3×M4(2) in GAP, Magma, Sage, TeX

C_5\times S_3\times M_4(2)
% in TeX

G:=Group("C5xS3xM4(2)");
// GroupNames label

G:=SmallGroup(480,785);
// by ID

G=gap.SmallGroup(480,785);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,891,226,102,15686]);
// Polycyclic

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

׿
×
𝔽