Copied to
clipboard

## G = C5×A4⋊C8order 480 = 25·3·5

### Direct product of C5 and A4⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C5×A4⋊C8
 Chief series C1 — C22 — A4 — C2×A4 — C4×A4 — A4×C20 — C5×A4⋊C8
 Lower central A4 — C5×A4⋊C8
 Upper central C1 — C20

Generators and relations for C5×A4⋊C8
G = < a,b,c,d,e | a5=b2=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Smallest permutation representation of C5×A4⋊C8
On 120 points
Generators in S120
(1 94 119 78 103)(2 95 120 79 104)(3 96 113 80 97)(4 89 114 73 98)(5 90 115 74 99)(6 91 116 75 100)(7 92 117 76 101)(8 93 118 77 102)(9 34 58 83 108)(10 35 59 84 109)(11 36 60 85 110)(12 37 61 86 111)(13 38 62 87 112)(14 39 63 88 105)(15 40 64 81 106)(16 33 57 82 107)(17 42 66 27 52)(18 43 67 28 53)(19 44 68 29 54)(20 45 69 30 55)(21 46 70 31 56)(22 47 71 32 49)(23 48 72 25 50)(24 41 65 26 51)
(2 6)(4 8)(9 13)(10 14)(11 15)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(43 47)(49 53)(51 55)(57 61)(58 62)(59 63)(60 64)(65 69)(67 71)(73 77)(75 79)(81 85)(82 86)(83 87)(84 88)(89 93)(91 95)(98 102)(100 104)(105 109)(106 110)(107 111)(108 112)(114 118)(116 120)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(113 117)(114 118)(115 119)(116 120)
(1 35 43)(2 44 36)(3 37 45)(4 46 38)(5 39 47)(6 48 40)(7 33 41)(8 42 34)(9 102 17)(10 18 103)(11 104 19)(12 20 97)(13 98 21)(14 22 99)(15 100 23)(16 24 101)(25 81 116)(26 117 82)(27 83 118)(28 119 84)(29 85 120)(30 113 86)(31 87 114)(32 115 88)(49 74 105)(50 106 75)(51 76 107)(52 108 77)(53 78 109)(54 110 79)(55 80 111)(56 112 73)(57 65 92)(58 93 66)(59 67 94)(60 95 68)(61 69 96)(62 89 70)(63 71 90)(64 91 72)
(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)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 5C 5D 6 8A ··· 8H 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 30A 30B 30C 30D 40A ··· 40AF 60A ··· 60H order 1 2 2 2 3 4 4 4 4 5 5 5 5 6 8 ··· 8 10 10 10 10 10 ··· 10 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 30 30 30 40 ··· 40 60 ··· 60 size 1 1 3 3 8 1 1 3 3 1 1 1 1 8 6 ··· 6 1 1 1 1 3 ··· 3 8 8 8 8 8 8 1 ··· 1 3 ··· 3 8 8 8 8 6 ··· 6 8 ··· 8

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 type + + + - + image C1 C2 C4 C5 C8 C10 C20 C40 S3 Dic3 C3⋊C8 C5×S3 C5×Dic3 C5×C3⋊C8 S4 A4⋊C4 A4⋊C8 C5×S4 C5×A4⋊C4 C5×A4⋊C8 kernel C5×A4⋊C8 A4×C20 C10×A4 A4⋊C8 C5×A4 C4×A4 C2×A4 A4 C22×C20 C22×C10 C2×C10 C22×C4 C23 C22 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 4 4 8 16 1 1 2 4 4 8 2 2 4 8 8 16

Matrix representation of C5×A4⋊C8 in GL3(𝔽241) generated by

 87 0 0 0 87 0 0 0 87
,
 1 0 0 1 240 0 1 0 240
,
 240 0 0 240 1 0 0 0 240
,
 1 0 239 0 0 240 0 1 240
,
 8 0 225 0 8 233 0 0 233
G:=sub<GL(3,GF(241))| [87,0,0,0,87,0,0,0,87],[1,1,1,0,240,0,0,0,240],[240,240,0,0,1,0,0,0,240],[1,0,0,0,0,1,239,240,240],[8,0,0,0,8,0,225,233,233] >;

C5×A4⋊C8 in GAP, Magma, Sage, TeX

C_5\times A_4\rtimes C_8
% in TeX

G:=Group("C5xA4:C8");
// GroupNames label

G:=SmallGroup(480,255);
// by ID

G=gap.SmallGroup(480,255);
# by ID

G:=PCGroup([7,-2,-5,-2,-2,-3,-2,2,70,58,2804,10085,285,5886,475]);
// Polycyclic

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

Export

׿
×
𝔽