Copied to
clipboard

## G = C3×C8⋊F5order 480 = 25·3·5

### Direct product of C3 and C8⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C8⋊F5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C12×F5 — C3×C8⋊F5
 Lower central C5 — C10 — C3×C8⋊F5
 Upper central C1 — C12 — C24

Generators and relations for C3×C8⋊F5
G = < a,b,c,d | a3=b8=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c3 >

Subgroups: 232 in 80 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×C12, C3×D5, C30, C8⋊C4, C52C8, C40, C5⋊C8, C4×D5, C2×F5, C4×C12, C2×C24, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C3×C8⋊C4, C3×C52C8, C120, C3×C5⋊C8, D5×C12, C6×F5, C8⋊F5, D5×C24, C3×D5⋊C8, C12×F5, C3×C8⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, M4(2), F5, C2×C12, C8⋊C4, C2×F5, C4×C12, C3×M4(2), C3×F5, C4×F5, C3×C8⋊C4, C6×F5, C8⋊F5, C12×F5, C3×C8⋊F5

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 6D 6E 6F 8A 8B 8C ··· 8H 10 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 15A 15B 20A 20B 24A 24B 24C 24D 24E ··· 24P 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 6 6 6 6 6 6 8 8 8 ··· 8 10 12 12 12 12 12 12 12 12 12 ··· 12 15 15 20 20 24 24 24 24 24 ··· 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 5 5 1 1 1 1 5 5 10 10 10 10 4 1 1 5 5 5 5 2 2 10 ··· 10 4 1 1 1 1 5 5 5 5 10 ··· 10 4 4 4 4 2 2 2 2 10 ··· 10 4 4 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 1 1 1 1 2 2 4 4 4 4 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C4 C4 C6 C6 C6 C12 C12 C12 C12 M4(2) C3×M4(2) F5 C2×F5 C3×F5 C4×F5 C6×F5 C8⋊F5 C12×F5 C3×C8⋊F5 kernel C3×C8⋊F5 D5×C24 C3×D5⋊C8 C12×F5 C8⋊F5 C3×C5⋊2C8 C120 C3×C5⋊C8 C6×F5 C8×D5 D5⋊C8 C4×F5 C5⋊2C8 C40 C5⋊C8 C2×F5 C3×D5 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 4 4 2 2 2 4 4 8 8 4 8 1 1 2 2 2 4 4 8

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

 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 240 128 0 0 0 0 152 1 0 0 0 0 0 0 177 0 0 0 0 0 0 177 0 0 0 0 0 0 177 0 0 0 0 0 0 177
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0 1 0 0 0 240 0 0 1 0 0 240 0 0 0
,
 64 0 0 0 0 0 1 177 0 0 0 0 0 0 1 0 240 0 0 0 0 0 240 1 0 0 0 1 240 0 0 0 0 0 240 0

G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[240,152,0,0,0,0,128,1,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[64,1,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,240,240,240,240,0,0,0,1,0,0] >;

C3×C8⋊F5 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes F_5
% in TeX

G:=Group("C3xC8:F5");
// GroupNames label

G:=SmallGroup(480,272);
// by ID

G=gap.SmallGroup(480,272);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,102,9414,1595]);
// Polycyclic

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

׿
×
𝔽