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## G = C3×D5⋊C8order 240 = 24·3·5

### Direct product of C3 and D5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×D5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C3×D5⋊C8
 Lower central C5 — C3×D5⋊C8
 Upper central C1 — C12

Generators and relations for C3×D5⋊C8
G = < a,b,c,d | a3=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Smallest permutation representation of C3×D5⋊C8
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)
(1 105 22 46 49)(2 47 106 50 23)(3 51 48 24 107)(4 17 52 108 41)(5 109 18 42 53)(6 43 110 54 19)(7 55 44 20 111)(8 21 56 112 45)(9 75 100 69 92)(10 70 76 93 101)(11 94 71 102 77)(12 103 95 78 72)(13 79 104 65 96)(14 66 80 89 97)(15 90 67 98 73)(16 99 91 74 68)(25 39 114 57 83)(26 58 40 84 115)(27 85 59 116 33)(28 117 86 34 60)(29 35 118 61 87)(30 62 36 88 119)(31 81 63 120 37)(32 113 82 38 64)
(1 49)(2 23)(3 107)(4 41)(5 53)(6 19)(7 111)(8 45)(9 69)(10 76)(12 103)(13 65)(14 80)(16 99)(17 108)(20 55)(21 112)(24 51)(26 58)(27 116)(28 86)(30 62)(31 120)(32 82)(34 60)(35 87)(36 119)(38 64)(39 83)(40 115)(42 109)(43 54)(46 105)(47 50)(57 114)(59 85)(61 118)(63 81)(67 98)(68 91)(71 102)(72 95)(73 90)(75 100)(77 94)(79 104)(89 97)(93 101)
(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)| (1,100,60)(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), (1,105,22,46,49)(2,47,106,50,23)(3,51,48,24,107)(4,17,52,108,41)(5,109,18,42,53)(6,43,110,54,19)(7,55,44,20,111)(8,21,56,112,45)(9,75,100,69,92)(10,70,76,93,101)(11,94,71,102,77)(12,103,95,78,72)(13,79,104,65,96)(14,66,80,89,97)(15,90,67,98,73)(16,99,91,74,68)(25,39,114,57,83)(26,58,40,84,115)(27,85,59,116,33)(28,117,86,34,60)(29,35,118,61,87)(30,62,36,88,119)(31,81,63,120,37)(32,113,82,38,64), (1,49)(2,23)(3,107)(4,41)(5,53)(6,19)(7,111)(8,45)(9,69)(10,76)(12,103)(13,65)(14,80)(16,99)(17,108)(20,55)(21,112)(24,51)(26,58)(27,116)(28,86)(30,62)(31,120)(32,82)(34,60)(35,87)(36,119)(38,64)(39,83)(40,115)(42,109)(43,54)(46,105)(47,50)(57,114)(59,85)(61,118)(63,81)(67,98)(68,91)(71,102)(72,95)(73,90)(75,100)(77,94)(79,104)(89,97)(93,101), (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( (1,100,60)(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), (1,105,22,46,49)(2,47,106,50,23)(3,51,48,24,107)(4,17,52,108,41)(5,109,18,42,53)(6,43,110,54,19)(7,55,44,20,111)(8,21,56,112,45)(9,75,100,69,92)(10,70,76,93,101)(11,94,71,102,77)(12,103,95,78,72)(13,79,104,65,96)(14,66,80,89,97)(15,90,67,98,73)(16,99,91,74,68)(25,39,114,57,83)(26,58,40,84,115)(27,85,59,116,33)(28,117,86,34,60)(29,35,118,61,87)(30,62,36,88,119)(31,81,63,120,37)(32,113,82,38,64), (1,49)(2,23)(3,107)(4,41)(5,53)(6,19)(7,111)(8,45)(9,69)(10,76)(12,103)(13,65)(14,80)(16,99)(17,108)(20,55)(21,112)(24,51)(26,58)(27,116)(28,86)(30,62)(31,120)(32,82)(34,60)(35,87)(36,119)(38,64)(39,83)(40,115)(42,109)(43,54)(46,105)(47,50)(57,114)(59,85)(61,118)(63,81)(67,98)(68,91)(71,102)(72,95)(73,90)(75,100)(77,94)(79,104)(89,97)(93,101), (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([[(1,100,60),(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)], [(1,105,22,46,49),(2,47,106,50,23),(3,51,48,24,107),(4,17,52,108,41),(5,109,18,42,53),(6,43,110,54,19),(7,55,44,20,111),(8,21,56,112,45),(9,75,100,69,92),(10,70,76,93,101),(11,94,71,102,77),(12,103,95,78,72),(13,79,104,65,96),(14,66,80,89,97),(15,90,67,98,73),(16,99,91,74,68),(25,39,114,57,83),(26,58,40,84,115),(27,85,59,116,33),(28,117,86,34,60),(29,35,118,61,87),(30,62,36,88,119),(31,81,63,120,37),(32,113,82,38,64)], [(1,49),(2,23),(3,107),(4,41),(5,53),(6,19),(7,111),(8,45),(9,69),(10,76),(12,103),(13,65),(14,80),(16,99),(17,108),(20,55),(21,112),(24,51),(26,58),(27,116),(28,86),(30,62),(31,120),(32,82),(34,60),(35,87),(36,119),(38,64),(39,83),(40,115),(42,109),(43,54),(46,105),(47,50),(57,114),(59,85),(61,118),(63,81),(67,98),(68,91),(71,102),(72,95),(73,90),(75,100),(77,94),(79,104),(89,97),(93,101)], [(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)]])

C3×D5⋊C8 is a maximal subgroup of
C30.C42  C30.4C42  D12⋊F5  Dic30⋊C4  F5×C24  D12.2F5  D60.C4  C5⋊C8⋊D6
C3×D5⋊C8 is a maximal quotient of
C12×C5⋊C8

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 F5 C2×F5 C3×F5 D5⋊C8 C6×F5 C3×D5⋊C8 kernel C3×D5⋊C8 C3×C5⋊C8 D5×C12 D5⋊C8 C60 C6×D5 C5⋊C8 C4×D5 C3×D5 C20 D10 D5 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 1 2 2 2 4

Matrix representation of C3×D5⋊C8 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 240 1 0 0 240 0 1 0 240 0 0 1 240 0 0 0
,
 240 0 0 0 240 0 0 1 240 0 1 0 240 1 0 0
,
 179 62 38 0 217 62 0 179 179 0 62 217 0 38 62 179
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[240,240,240,240,1,0,0,0,0,1,0,0,0,0,1,0],[240,240,240,240,0,0,0,1,0,0,1,0,0,1,0,0],[179,217,179,0,62,62,0,38,38,0,62,62,0,179,217,179] >;

C3×D5⋊C8 in GAP, Magma, Sage, TeX

C_3\times D_5\rtimes C_8
% in TeX

G:=Group("C3xD5:C8");
// GroupNames label

G:=SmallGroup(240,111);
// by ID

G=gap.SmallGroup(240,111);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,151,69,3461,599]);
// Polycyclic

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

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