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G = C5×C3⋊C8order 120 = 23·3·5

Direct product of C5 and C3⋊C8

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

Aliases: C5×C3⋊C8, C3⋊C40, C155C8, C6.C20, C30.5C4, C60.6C2, C20.4S3, C12.2C10, C10.3Dic3, C4.2(C5×S3), C2.(C5×Dic3), SmallGroup(120,1)

Series: Derived Chief Lower central Upper central

C1C3 — C5×C3⋊C8
C1C3C6C12C60 — C5×C3⋊C8
C3 — C5×C3⋊C8
C1C20

Generators and relations for C5×C3⋊C8
 G = < a,b,c | a5=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

3C8
3C40

Smallest permutation representation of C5×C3⋊C8
Regular action on 120 points
Generators in S120
(1 15 26 51 77)(2 16 27 52 78)(3 9 28 53 79)(4 10 29 54 80)(5 11 30 55 73)(6 12 31 56 74)(7 13 32 49 75)(8 14 25 50 76)(17 36 63 85 110)(18 37 64 86 111)(19 38 57 87 112)(20 39 58 88 105)(21 40 59 81 106)(22 33 60 82 107)(23 34 61 83 108)(24 35 62 84 109)(41 103 69 94 119)(42 104 70 95 120)(43 97 71 96 113)(44 98 72 89 114)(45 99 65 90 115)(46 100 66 91 116)(47 101 67 92 117)(48 102 68 93 118)
(1 21 103)(2 104 22)(3 23 97)(4 98 24)(5 17 99)(6 100 18)(7 19 101)(8 102 20)(9 34 71)(10 72 35)(11 36 65)(12 66 37)(13 38 67)(14 68 39)(15 40 69)(16 70 33)(25 93 58)(26 59 94)(27 95 60)(28 61 96)(29 89 62)(30 63 90)(31 91 64)(32 57 92)(41 77 106)(42 107 78)(43 79 108)(44 109 80)(45 73 110)(46 111 74)(47 75 112)(48 105 76)(49 87 117)(50 118 88)(51 81 119)(52 120 82)(53 83 113)(54 114 84)(55 85 115)(56 116 86)
(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)| (1,15,26,51,77)(2,16,27,52,78)(3,9,28,53,79)(4,10,29,54,80)(5,11,30,55,73)(6,12,31,56,74)(7,13,32,49,75)(8,14,25,50,76)(17,36,63,85,110)(18,37,64,86,111)(19,38,57,87,112)(20,39,58,88,105)(21,40,59,81,106)(22,33,60,82,107)(23,34,61,83,108)(24,35,62,84,109)(41,103,69,94,119)(42,104,70,95,120)(43,97,71,96,113)(44,98,72,89,114)(45,99,65,90,115)(46,100,66,91,116)(47,101,67,92,117)(48,102,68,93,118), (1,21,103)(2,104,22)(3,23,97)(4,98,24)(5,17,99)(6,100,18)(7,19,101)(8,102,20)(9,34,71)(10,72,35)(11,36,65)(12,66,37)(13,38,67)(14,68,39)(15,40,69)(16,70,33)(25,93,58)(26,59,94)(27,95,60)(28,61,96)(29,89,62)(30,63,90)(31,91,64)(32,57,92)(41,77,106)(42,107,78)(43,79,108)(44,109,80)(45,73,110)(46,111,74)(47,75,112)(48,105,76)(49,87,117)(50,118,88)(51,81,119)(52,120,82)(53,83,113)(54,114,84)(55,85,115)(56,116,86), (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( (1,15,26,51,77)(2,16,27,52,78)(3,9,28,53,79)(4,10,29,54,80)(5,11,30,55,73)(6,12,31,56,74)(7,13,32,49,75)(8,14,25,50,76)(17,36,63,85,110)(18,37,64,86,111)(19,38,57,87,112)(20,39,58,88,105)(21,40,59,81,106)(22,33,60,82,107)(23,34,61,83,108)(24,35,62,84,109)(41,103,69,94,119)(42,104,70,95,120)(43,97,71,96,113)(44,98,72,89,114)(45,99,65,90,115)(46,100,66,91,116)(47,101,67,92,117)(48,102,68,93,118), (1,21,103)(2,104,22)(3,23,97)(4,98,24)(5,17,99)(6,100,18)(7,19,101)(8,102,20)(9,34,71)(10,72,35)(11,36,65)(12,66,37)(13,38,67)(14,68,39)(15,40,69)(16,70,33)(25,93,58)(26,59,94)(27,95,60)(28,61,96)(29,89,62)(30,63,90)(31,91,64)(32,57,92)(41,77,106)(42,107,78)(43,79,108)(44,109,80)(45,73,110)(46,111,74)(47,75,112)(48,105,76)(49,87,117)(50,118,88)(51,81,119)(52,120,82)(53,83,113)(54,114,84)(55,85,115)(56,116,86), (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([[(1,15,26,51,77),(2,16,27,52,78),(3,9,28,53,79),(4,10,29,54,80),(5,11,30,55,73),(6,12,31,56,74),(7,13,32,49,75),(8,14,25,50,76),(17,36,63,85,110),(18,37,64,86,111),(19,38,57,87,112),(20,39,58,88,105),(21,40,59,81,106),(22,33,60,82,107),(23,34,61,83,108),(24,35,62,84,109),(41,103,69,94,119),(42,104,70,95,120),(43,97,71,96,113),(44,98,72,89,114),(45,99,65,90,115),(46,100,66,91,116),(47,101,67,92,117),(48,102,68,93,118)], [(1,21,103),(2,104,22),(3,23,97),(4,98,24),(5,17,99),(6,100,18),(7,19,101),(8,102,20),(9,34,71),(10,72,35),(11,36,65),(12,66,37),(13,38,67),(14,68,39),(15,40,69),(16,70,33),(25,93,58),(26,59,94),(27,95,60),(28,61,96),(29,89,62),(30,63,90),(31,91,64),(32,57,92),(41,77,106),(42,107,78),(43,79,108),(44,109,80),(45,73,110),(46,111,74),(47,75,112),(48,105,76),(49,87,117),(50,118,88),(51,81,119),(52,120,82),(53,83,113),(54,114,84),(55,85,115),(56,116,86)], [(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)]])

C5×C3⋊C8 is a maximal subgroup of
D152C8  C20.32D6  D30.5C4  C3⋊D40  C6.D20  C15⋊SD16  C3⋊Dic20  S3×C40

60 conjugacy classes

class 1  2  3 4A4B5A5B5C5D 6 8A8B8C8D10A10B10C10D12A12B15A15B15C15D20A···20H30A30B30C30D40A···40P60A···60H
order123445555688881010101012121515151520···203030303040···4060···60
size1121111112333311112222221···122223···32···2

60 irreducible representations

dim11111111222222
type+++-
imageC1C2C4C5C8C10C20C40S3Dic3C3⋊C8C5×S3C5×Dic3C5×C3⋊C8
kernelC5×C3⋊C8C60C30C3⋊C8C15C12C6C3C20C10C5C4C2C1
# reps112444816112448

Matrix representation of C5×C3⋊C8 in GL2(𝔽41) generated by

160
016
,
037
3140
,
329
038
G:=sub<GL(2,GF(41))| [16,0,0,16],[0,31,37,40],[3,0,29,38] >;

C5×C3⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes C_8
% in TeX

G:=Group("C5xC3:C8");
// GroupNames label

G:=SmallGroup(120,1);
// by ID

G=gap.SmallGroup(120,1);
# by ID

G:=PCGroup([5,-2,-5,-2,-2,-3,50,42,2004]);
// Polycyclic

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

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Subgroup lattice of C5×C3⋊C8 in TeX

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