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## G = C3×C5⋊2C8order 120 = 23·3·5

### Direct product of C3 and C5⋊2C8

Aliases: C3×C52C8, C154C8, C52C24, C20.2C6, C60.5C2, C30.4C4, C12.4D5, C10.2C12, C6.2Dic5, C4.2(C3×D5), C2.(C3×Dic5), SmallGroup(120,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C60 — C3×C5⋊2C8
 Lower central C5 — C3×C5⋊2C8
 Upper central C1 — C12

Generators and relations for C3×C52C8
G = < a,b,c | a3=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×C52C8
Regular action 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 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 94 119)(18 95 120)(19 96 113)(20 89 114)(21 90 115)(22 91 116)(23 92 117)(24 93 118)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 41 70)(30 42 71)(31 43 72)(32 44 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)
(1 53 23 46 109)(2 110 47 24 54)(3 55 17 48 111)(4 112 41 18 56)(5 49 19 42 105)(6 106 43 20 50)(7 51 21 44 107)(8 108 45 22 52)(9 90 65 73 98)(10 99 74 66 91)(11 92 67 75 100)(12 101 76 68 93)(13 94 69 77 102)(14 103 78 70 95)(15 96 71 79 104)(16 97 80 72 89)(25 116 87 59 33)(26 34 60 88 117)(27 118 81 61 35)(28 36 62 82 119)(29 120 83 63 37)(30 38 64 84 113)(31 114 85 57 39)(32 40 58 86 115)
(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)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (1,53,23,46,109)(2,110,47,24,54)(3,55,17,48,111)(4,112,41,18,56)(5,49,19,42,105)(6,106,43,20,50)(7,51,21,44,107)(8,108,45,22,52)(9,90,65,73,98)(10,99,74,66,91)(11,92,67,75,100)(12,101,76,68,93)(13,94,69,77,102)(14,103,78,70,95)(15,96,71,79,104)(16,97,80,72,89)(25,116,87,59,33)(26,34,60,88,117)(27,118,81,61,35)(28,36,62,82,119)(29,120,83,63,37)(30,38,64,84,113)(31,114,85,57,39)(32,40,58,86,115), (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)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (1,53,23,46,109)(2,110,47,24,54)(3,55,17,48,111)(4,112,41,18,56)(5,49,19,42,105)(6,106,43,20,50)(7,51,21,44,107)(8,108,45,22,52)(9,90,65,73,98)(10,99,74,66,91)(11,92,67,75,100)(12,101,76,68,93)(13,94,69,77,102)(14,103,78,70,95)(15,96,71,79,104)(16,97,80,72,89)(25,116,87,59,33)(26,34,60,88,117)(27,118,81,61,35)(28,36,62,82,119)(29,120,83,63,37)(30,38,64,84,113)(31,114,85,57,39)(32,40,58,86,115), (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),(9,86,51),(10,87,52),(11,88,53),(12,81,54),(13,82,55),(14,83,56),(15,84,49),(16,85,50),(17,94,119),(18,95,120),(19,96,113),(20,89,114),(21,90,115),(22,91,116),(23,92,117),(24,93,118),(25,45,66),(26,46,67),(27,47,68),(28,48,69),(29,41,70),(30,42,71),(31,43,72),(32,44,65),(33,108,74),(34,109,75),(35,110,76),(36,111,77),(37,112,78),(38,105,79),(39,106,80),(40,107,73)], [(1,53,23,46,109),(2,110,47,24,54),(3,55,17,48,111),(4,112,41,18,56),(5,49,19,42,105),(6,106,43,20,50),(7,51,21,44,107),(8,108,45,22,52),(9,90,65,73,98),(10,99,74,66,91),(11,92,67,75,100),(12,101,76,68,93),(13,94,69,77,102),(14,103,78,70,95),(15,96,71,79,104),(16,97,80,72,89),(25,116,87,59,33),(26,34,60,88,117),(27,118,81,61,35),(28,36,62,82,119),(29,120,83,63,37),(30,38,64,84,113),(31,114,85,57,39),(32,40,58,86,115)], [(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×C52C8 is a maximal subgroup of
C15⋊C16  D152C8  D6.Dic5  D30.5C4  C5⋊D24  D12.D5  Dic6⋊D5  C5⋊Dic12  D5×C24  SL2(𝔽3).Dic5

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C24 D5 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C3×C5⋊2C8 kernel C3×C5⋊2C8 C60 C5⋊2C8 C30 C20 C15 C10 C5 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 2 2 4 4 4 8

Matrix representation of C3×C52C8 in GL3(𝔽241) generated by

 225 0 0 0 1 0 0 0 1
,
 1 0 0 0 51 240 0 1 0
,
 240 0 0 0 102 231 0 131 139
G:=sub<GL(3,GF(241))| [225,0,0,0,1,0,0,0,1],[1,0,0,0,51,1,0,240,0],[240,0,0,0,102,131,0,231,139] >;

C3×C52C8 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes_2C_8
% in TeX

G:=Group("C3xC5:2C8");
// GroupNames label

G:=SmallGroup(120,2);
// by ID

G=gap.SmallGroup(120,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-5,30,42,2404]);
// Polycyclic

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

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