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## G = F5×C3⋊C8order 480 = 25·3·5

### Direct product of F5 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for F5×C3⋊C8
G = < a,b,c,d | a5=b4=c3=d8=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 308 in 88 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C3⋊C8, C2×C12, C3×D5, C30, C4×C8, C52C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×C12, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C4×C3⋊C8, C5×C3⋊C8, C153C8, C15⋊C8, D5×C12, C6×F5, C8×F5, D5×C3⋊C8, C12×F5, C60.C4, F5×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, F5, C3⋊C8, C4×S3, C2×Dic3, C4×C8, C2×F5, C2×C3⋊C8, C4×Dic3, C4×F5, C4×C3⋊C8, S3×F5, C8×F5, Dic3×F5, F5×C3⋊C8

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 8 type + + + + + + - + + + - image C1 C2 C2 C2 C4 C4 C4 C4 C8 S3 D6 Dic3 C4×S3 C4×S3 C3⋊C8 F5 C2×F5 C4×F5 C8×F5 S3×F5 Dic3×F5 F5×C3⋊C8 kernel F5×C3⋊C8 D5×C3⋊C8 C12×F5 C60.C4 C5×C3⋊C8 C15⋊3C8 C15⋊C8 C6×F5 C3×F5 C4×F5 C4×D5 C2×F5 Dic5 C20 F5 C3⋊C8 C12 C6 C3 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 1 2 2 2 8 1 1 2 4 1 1 2

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

 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
,
 1 0 0 0 0 0 0 1 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
,
 0 240 0 0 0 0 1 240 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
,
 186 78 0 0 0 0 23 55 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8

G:=sub<GL(6,GF(241))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[186,23,0,0,0,0,78,55,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

F5×C3⋊C8 in GAP, Magma, Sage, TeX

F_5\times C_3\rtimes C_8
% in TeX

G:=Group("F5xC3:C8");
// GroupNames label

G:=SmallGroup(480,223);
// by ID

G=gap.SmallGroup(480,223);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,80,1356,9414,4724]);
// Polycyclic

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

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