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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 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 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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