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## G = C8×C3⋊F5order 480 = 25·3·5

### Direct product of C8 and C3⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C8×C3⋊F5
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C4×C3⋊F5 — C8×C3⋊F5
 Lower central C15 — C8×C3⋊F5
 Upper central C1 — C8

Generators and relations for C8×C3⋊F5
G = < a,b,c,d | a8=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 364 in 88 conjugacy classes, 41 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C24, C24, C2×Dic3, C2×C12, C3×D5, C30, C4×C8, C52C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C8×Dic3, C3×C52C8, C120, C15⋊C8, D5×C12, C2×C3⋊F5, C8×F5, D5×C24, C60.C4, C4×C3⋊F5, C8×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, F5, C4×S3, C2×Dic3, C4×C8, C2×F5, S3×C8, C4×Dic3, C3⋊F5, C4×F5, C8×Dic3, C2×C3⋊F5, C8×F5, C4×C3⋊F5, C8×C3⋊F5

Smallest permutation representation of C8×C3⋊F5
On 120 points
Generators in S120
(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 64 50)(2 57 51)(3 58 52)(4 59 53)(5 60 54)(6 61 55)(7 62 56)(8 63 49)(9 69 39)(10 70 40)(11 71 33)(12 72 34)(13 65 35)(14 66 36)(15 67 37)(16 68 38)(17 48 120)(18 41 113)(19 42 114)(20 43 115)(21 44 116)(22 45 117)(23 46 118)(24 47 119)(25 95 102)(26 96 103)(27 89 104)(28 90 97)(29 91 98)(30 92 99)(31 93 100)(32 94 101)(73 84 105)(74 85 106)(75 86 107)(76 87 108)(77 88 109)(78 81 110)(79 82 111)(80 83 112)
(1 73 113 65 99)(2 74 114 66 100)(3 75 115 67 101)(4 76 116 68 102)(5 77 117 69 103)(6 78 118 70 104)(7 79 119 71 97)(8 80 120 72 98)(9 96 54 109 45)(10 89 55 110 46)(11 90 56 111 47)(12 91 49 112 48)(13 92 50 105 41)(14 93 51 106 42)(15 94 52 107 43)(16 95 53 108 44)(17 34 29 63 83)(18 35 30 64 84)(19 36 31 57 85)(20 37 32 58 86)(21 38 25 59 87)(22 39 26 60 88)(23 40 27 61 81)(24 33 28 62 82)
(9 88 45 26)(10 81 46 27)(11 82 47 28)(12 83 48 29)(13 84 41 30)(14 85 42 31)(15 86 43 32)(16 87 44 25)(17 91 34 112)(18 92 35 105)(19 93 36 106)(20 94 37 107)(21 95 38 108)(22 96 39 109)(23 89 40 110)(24 90 33 111)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(65 73 113 99)(66 74 114 100)(67 75 115 101)(68 76 116 102)(69 77 117 103)(70 78 118 104)(71 79 119 97)(72 80 120 98)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - - + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 S3 Dic3 Dic3 D6 C4×S3 C4×S3 S3×C8 F5 C2×F5 C3⋊F5 C4×F5 C2×C3⋊F5 C8×F5 C4×C3⋊F5 C8×C3⋊F5 kernel C8×C3⋊F5 D5×C24 C60.C4 C4×C3⋊F5 C3×C5⋊2C8 C120 C15⋊C8 C2×C3⋊F5 C3⋊F5 C8×D5 C5⋊2C8 C40 C4×D5 Dic5 D10 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 1 1 1 2 2 8 1 1 2 2 2 4 4 8

Matrix representation of C8×C3⋊F5 in GL6(𝔽241)

 211 0 0 0 0 0 0 211 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 240 240 0 0 0 0 1 0 0 0 0 0 0 0 126 0 12 12 0 0 229 114 229 0 0 0 0 229 114 229 0 0 12 12 0 126
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 177 0 0 0 0 0 64 64 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,0,211,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,126,229,0,12,0,0,0,114,229,12,0,0,12,229,114,0,0,0,12,0,229,126],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[177,64,0,0,0,0,0,64,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C8×C3⋊F5 in GAP, Magma, Sage, TeX

C_8\times C_3\rtimes F_5
% in TeX

G:=Group("C8xC3:F5");
// GroupNames label

G:=SmallGroup(480,296);
// by ID

G=gap.SmallGroup(480,296);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,80,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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