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G = Q8×C15order 120 = 23·3·5

Direct product of C15 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C15, C4.C30, C60.7C2, C20.3C6, C12.3C10, C30.24C22, C10.7(C2×C6), C2.2(C2×C30), C6.7(C2×C10), SmallGroup(120,33)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C15
C1C2C10C30C60 — Q8×C15
C1C2 — Q8×C15
C1C30 — Q8×C15

Generators and relations for Q8×C15
 G = < a,b,c | a15=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C15
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)
(1 106 18 31)(2 107 19 32)(3 108 20 33)(4 109 21 34)(5 110 22 35)(6 111 23 36)(7 112 24 37)(8 113 25 38)(9 114 26 39)(10 115 27 40)(11 116 28 41)(12 117 29 42)(13 118 30 43)(14 119 16 44)(15 120 17 45)(46 62 98 82)(47 63 99 83)(48 64 100 84)(49 65 101 85)(50 66 102 86)(51 67 103 87)(52 68 104 88)(53 69 105 89)(54 70 91 90)(55 71 92 76)(56 72 93 77)(57 73 94 78)(58 74 95 79)(59 75 96 80)(60 61 97 81)
(1 76 18 71)(2 77 19 72)(3 78 20 73)(4 79 21 74)(5 80 22 75)(6 81 23 61)(7 82 24 62)(8 83 25 63)(9 84 26 64)(10 85 27 65)(11 86 28 66)(12 87 29 67)(13 88 30 68)(14 89 16 69)(15 90 17 70)(31 55 106 92)(32 56 107 93)(33 57 108 94)(34 58 109 95)(35 59 110 96)(36 60 111 97)(37 46 112 98)(38 47 113 99)(39 48 114 100)(40 49 115 101)(41 50 116 102)(42 51 117 103)(43 52 118 104)(44 53 119 105)(45 54 120 91)

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

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

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

Q8×C15 is a maximal subgroup of   Q82D15  C157Q16  Q83D15

75 conjugacy classes

class 1  2 3A3B4A4B4C5A5B5C5D6A6B10A10B10C10D12A···12F15A···15H20A···20L30A···30H60A···60X
order12334445555661010101012···1215···1520···2030···3060···60
size111122211111111112···21···12···21···12···2

75 irreducible representations

dim111111112222
type++-
imageC1C2C3C5C6C10C15C30Q8C3×Q8C5×Q8Q8×C15
kernelQ8×C15C60C5×Q8C3×Q8C20C12Q8C4C15C5C3C1
# reps13246128241248

Matrix representation of Q8×C15 in GL2(𝔽31) generated by

200
020
,
2717
304
,
018
120
G:=sub<GL(2,GF(31))| [20,0,0,20],[27,30,17,4],[0,12,18,0] >;

Q8×C15 in GAP, Magma, Sage, TeX

Q_8\times C_{15}
% in TeX

G:=Group("Q8xC15");
// GroupNames label

G:=SmallGroup(120,33);
// by ID

G=gap.SmallGroup(120,33);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-2,300,621,306]);
// Polycyclic

G:=Group<a,b,c|a^15=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C15 in TeX

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