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

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

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

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

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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