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G = C3×C52C8order 120 = 23·3·5

Direct product of C3 and C52C8

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

C1C5 — C3×C52C8
C1C5C10C20C60 — C3×C52C8
C5 — C3×C52C8
C1C12

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

5C8
5C24

Smallest permutation representation of C3×C52C8
Regular action on 120 points
Generators in S120
(1 100 65)(2 101 66)(3 102 67)(4 103 68)(5 104 69)(6 97 70)(7 98 71)(8 99 72)(9 86 43)(10 87 44)(11 88 45)(12 81 46)(13 82 47)(14 83 48)(15 84 41)(16 85 42)(17 55 58)(18 56 59)(19 49 60)(20 50 61)(21 51 62)(22 52 63)(23 53 64)(24 54 57)(25 90 115)(26 91 116)(27 92 117)(28 93 118)(29 94 119)(30 95 120)(31 96 113)(32 89 114)(33 111 74)(34 112 75)(35 105 76)(36 106 77)(37 107 78)(38 108 79)(39 109 80)(40 110 73)
(1 48 27 51 112)(2 105 52 28 41)(3 42 29 53 106)(4 107 54 30 43)(5 44 31 55 108)(6 109 56 32 45)(7 46 25 49 110)(8 111 50 26 47)(9 103 78 57 95)(10 96 58 79 104)(11 97 80 59 89)(12 90 60 73 98)(13 99 74 61 91)(14 92 62 75 100)(15 101 76 63 93)(16 94 64 77 102)(17 38 69 87 113)(18 114 88 70 39)(19 40 71 81 115)(20 116 82 72 33)(21 34 65 83 117)(22 118 84 66 35)(23 36 67 85 119)(24 120 86 68 37)
(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)| (1,100,65)(2,101,66)(3,102,67)(4,103,68)(5,104,69)(6,97,70)(7,98,71)(8,99,72)(9,86,43)(10,87,44)(11,88,45)(12,81,46)(13,82,47)(14,83,48)(15,84,41)(16,85,42)(17,55,58)(18,56,59)(19,49,60)(20,50,61)(21,51,62)(22,52,63)(23,53,64)(24,54,57)(25,90,115)(26,91,116)(27,92,117)(28,93,118)(29,94,119)(30,95,120)(31,96,113)(32,89,114)(33,111,74)(34,112,75)(35,105,76)(36,106,77)(37,107,78)(38,108,79)(39,109,80)(40,110,73), (1,48,27,51,112)(2,105,52,28,41)(3,42,29,53,106)(4,107,54,30,43)(5,44,31,55,108)(6,109,56,32,45)(7,46,25,49,110)(8,111,50,26,47)(9,103,78,57,95)(10,96,58,79,104)(11,97,80,59,89)(12,90,60,73,98)(13,99,74,61,91)(14,92,62,75,100)(15,101,76,63,93)(16,94,64,77,102)(17,38,69,87,113)(18,114,88,70,39)(19,40,71,81,115)(20,116,82,72,33)(21,34,65,83,117)(22,118,84,66,35)(23,36,67,85,119)(24,120,86,68,37), (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( (1,100,65)(2,101,66)(3,102,67)(4,103,68)(5,104,69)(6,97,70)(7,98,71)(8,99,72)(9,86,43)(10,87,44)(11,88,45)(12,81,46)(13,82,47)(14,83,48)(15,84,41)(16,85,42)(17,55,58)(18,56,59)(19,49,60)(20,50,61)(21,51,62)(22,52,63)(23,53,64)(24,54,57)(25,90,115)(26,91,116)(27,92,117)(28,93,118)(29,94,119)(30,95,120)(31,96,113)(32,89,114)(33,111,74)(34,112,75)(35,105,76)(36,106,77)(37,107,78)(38,108,79)(39,109,80)(40,110,73), (1,48,27,51,112)(2,105,52,28,41)(3,42,29,53,106)(4,107,54,30,43)(5,44,31,55,108)(6,109,56,32,45)(7,46,25,49,110)(8,111,50,26,47)(9,103,78,57,95)(10,96,58,79,104)(11,97,80,59,89)(12,90,60,73,98)(13,99,74,61,91)(14,92,62,75,100)(15,101,76,63,93)(16,94,64,77,102)(17,38,69,87,113)(18,114,88,70,39)(19,40,71,81,115)(20,116,82,72,33)(21,34,65,83,117)(22,118,84,66,35)(23,36,67,85,119)(24,120,86,68,37), (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([[(1,100,65),(2,101,66),(3,102,67),(4,103,68),(5,104,69),(6,97,70),(7,98,71),(8,99,72),(9,86,43),(10,87,44),(11,88,45),(12,81,46),(13,82,47),(14,83,48),(15,84,41),(16,85,42),(17,55,58),(18,56,59),(19,49,60),(20,50,61),(21,51,62),(22,52,63),(23,53,64),(24,54,57),(25,90,115),(26,91,116),(27,92,117),(28,93,118),(29,94,119),(30,95,120),(31,96,113),(32,89,114),(33,111,74),(34,112,75),(35,105,76),(36,106,77),(37,107,78),(38,108,79),(39,109,80),(40,110,73)], [(1,48,27,51,112),(2,105,52,28,41),(3,42,29,53,106),(4,107,54,30,43),(5,44,31,55,108),(6,109,56,32,45),(7,46,25,49,110),(8,111,50,26,47),(9,103,78,57,95),(10,96,58,79,104),(11,97,80,59,89),(12,90,60,73,98),(13,99,74,61,91),(14,92,62,75,100),(15,101,76,63,93),(16,94,64,77,102),(17,38,69,87,113),(18,114,88,70,39),(19,40,71,81,115),(20,116,82,72,33),(21,34,65,83,117),(22,118,84,66,35),(23,36,67,85,119),(24,120,86,68,37)], [(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 3A3B4A4B5A5B6A6B8A8B8C8D10A10B12A12B12C12D15A15B15C15D20A20B20C20D24A···24H30A30B30C30D60A···60H
order12334455668888101012121212151515152020202024···243030303060···60
size11111122115555221111222222225···522222···2

48 irreducible representations

dim11111111222222
type+++-
imageC1C2C3C4C6C8C12C24D5Dic5C3×D5C52C8C3×Dic5C3×C52C8
kernelC3×C52C8C60C52C8C30C20C15C10C5C12C6C4C3C2C1
# reps11222448224448

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

22500
010
001
,
100
051240
010
,
24000
0102231
0131139
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

Export

Subgroup lattice of C3×C52C8 in TeX

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