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G = C153C8order 120 = 23·3·5

1st semidirect product of C15 and C8 acting via C8/C4=C2

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

Aliases: C153C8, C6.Dic5, C60.2C2, C30.3C4, C20.2S3, C12.2D5, C4.2D15, C2.Dic15, C10.2Dic3, C3⋊(C52C8), C52(C3⋊C8), SmallGroup(120,3)

Series: Derived Chief Lower central Upper central

C1C15 — C153C8
C1C5C15C30C60 — C153C8
C15 — C153C8
C1C4

Generators and relations for C153C8
 G = < a,b | a15=b8=1, bab-1=a-1 >

15C8
5C3⋊C8
3C52C8

Smallest permutation representation of C153C8
Regular action on 120 points
Generators in S120
(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)
(1 117 47 76 17 99 39 75)(2 116 48 90 18 98 40 74)(3 115 49 89 19 97 41 73)(4 114 50 88 20 96 42 72)(5 113 51 87 21 95 43 71)(6 112 52 86 22 94 44 70)(7 111 53 85 23 93 45 69)(8 110 54 84 24 92 31 68)(9 109 55 83 25 91 32 67)(10 108 56 82 26 105 33 66)(11 107 57 81 27 104 34 65)(12 106 58 80 28 103 35 64)(13 120 59 79 29 102 36 63)(14 119 60 78 30 101 37 62)(15 118 46 77 16 100 38 61)

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

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

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

C153C8 is a maximal subgroup of
D5×C3⋊C8  S3×C52C8  C20.32D6  D6.Dic5  C15⋊D8  C30.D4  C20.D6  C15⋊Q16  C8×D15  C40⋊S3  C60.7C4  D4⋊D15  D4.D15  Q82D15  C157Q16  C453C8  C60.S3  C20.S4  C52U2(𝔽3)
C153C8 is a maximal quotient of
C153C16  C453C8  C60.S3  C20.S4

36 conjugacy classes

class 1  2  3 4A4B5A5B 6 8A8B8C8D10A10B12A12B15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order1234455688881010121215151515202020203030303060···60
size112112221515151522222222222222222···2

36 irreducible representations

dim1111222222222
type++++--+-
imageC1C2C4C8S3D5Dic3Dic5C3⋊C8D15C52C8Dic15C153C8
kernelC153C8C60C30C15C20C12C10C6C5C4C3C2C1
# reps1124121224448

Matrix representation of C153C8 in GL2(𝔽29) generated by

117
2813
,
017
10
G:=sub<GL(2,GF(29))| [1,28,17,13],[0,1,17,0] >;

C153C8 in GAP, Magma, Sage, TeX

C_{15}\rtimes_3C_8
% in TeX

G:=Group("C15:3C8");
// GroupNames label

G:=SmallGroup(120,3);
// by ID

G=gap.SmallGroup(120,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,10,26,323,2404]);
// Polycyclic

G:=Group<a,b|a^15=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C153C8 in TeX

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