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G = C6xQ16order 96 = 25·3

Direct product of C6 and Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C6xQ16, C12.43D4, C24.22C22, C12.46C23, (C2xC8).4C6, C8.5(C2xC6), C4.8(C3xD4), C2.13(C6xD4), C6.76(C2xD4), (C2xC6).54D4, Q8.4(C2xC6), (C2xQ8).6C6, (C6xQ8).9C2, (C2xC24).10C2, C4.3(C22xC6), C22.16(C3xD4), (C3xQ8).12C22, (C2xC12).131C22, (C2xC4).27(C2xC6), SmallGroup(96,181)

Series: Derived Chief Lower central Upper central

C1C4 — C6xQ16
C1C2C4C12C3xQ8C3xQ16 — C6xQ16
C1C2C4 — C6xQ16
C1C2xC6C2xC12 — C6xQ16

Generators and relations for C6xQ16
 G = < a,b,c | a6=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 60 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2xC4, C2xC4, Q8, Q8, C12, C12, C2xC6, C2xC8, Q16, C2xQ8, C24, C2xC12, C2xC12, C3xQ8, C3xQ8, C2xQ16, C2xC24, C3xQ16, C6xQ8, C6xQ16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, Q16, C2xD4, C3xD4, C22xC6, C2xQ16, C3xQ16, C6xD4, C6xQ16

Smallest permutation representation of C6xQ16
Regular action on 96 points
Generators in S96
(1 28 55 95 19 74)(2 29 56 96 20 75)(3 30 49 89 21 76)(4 31 50 90 22 77)(5 32 51 91 23 78)(6 25 52 92 24 79)(7 26 53 93 17 80)(8 27 54 94 18 73)(9 69 81 63 37 46)(10 70 82 64 38 47)(11 71 83 57 39 48)(12 72 84 58 40 41)(13 65 85 59 33 42)(14 66 86 60 34 43)(15 67 87 61 35 44)(16 68 88 62 36 45)
(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)
(1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(25 70 29 66)(26 69 30 65)(27 68 31 72)(28 67 32 71)(41 73 45 77)(42 80 46 76)(43 79 47 75)(44 78 48 74)(49 85 53 81)(50 84 54 88)(51 83 55 87)(52 82 56 86)(57 95 61 91)(58 94 62 90)(59 93 63 89)(60 92 64 96)

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

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

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

C6xQ16 is a maximal subgroup of
C6.5Q32  Q16.Dic3  C24.27C23  Dic3:3Q16  C24.26D4  Q16:Dic3  (C2xQ16):S3  D6:5Q16  D12.17D4  D6:3Q16  C24.36D4  C24.37D4  C24.28D4  C24.29D4  D12.30D4

42 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F6A···6F8A8B8C8D12A12B12C12D12E···12L24A···24H
order1222334444446···688881212121212···1224···24
size1111112244441···1222222224···42···2

42 irreducible representations

dim11111111222222
type++++++-
imageC1C2C2C2C3C6C6C6D4D4Q16C3xD4C3xD4C3xQ16
kernelC6xQ16C2xC24C3xQ16C6xQ8C2xQ16C2xC8Q16C2xQ8C12C2xC6C6C4C22C2
# reps11422284114228

Matrix representation of C6xQ16 in GL3(F73) generated by

900
0720
0072
,
100
05716
05757
,
7200
02154
05452
G:=sub<GL(3,GF(73))| [9,0,0,0,72,0,0,0,72],[1,0,0,0,57,57,0,16,57],[72,0,0,0,21,54,0,54,52] >;

C6xQ16 in GAP, Magma, Sage, TeX

C_6\times Q_{16}
% in TeX

G:=Group("C6xQ16");
// GroupNames label

G:=SmallGroup(96,181);
// by ID

G=gap.SmallGroup(96,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,295,2164,1090,88]);
// Polycyclic

G:=Group<a,b,c|a^6=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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