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G = C7×Q16order 112 = 24·7

Direct product of C7 and Q16

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

Aliases: C7×Q16, C8.C14, Q8.C14, C56.3C2, C14.16D4, C28.19C22, C2.5(C7×D4), C4.3(C2×C14), (C7×Q8).2C2, SmallGroup(112,26)

Series: Derived Chief Lower central Upper central

C1C4 — C7×Q16
C1C2C4C28C7×Q8 — C7×Q16
C1C2C4 — C7×Q16
C1C14C28 — C7×Q16

Generators and relations for C7×Q16
 G = < a,b,c | a7=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C28
2C28

Smallest permutation representation of C7×Q16
Regular action on 112 points
Generators in S112
(1 82 103 107 37 31 57)(2 83 104 108 38 32 58)(3 84 97 109 39 25 59)(4 85 98 110 40 26 60)(5 86 99 111 33 27 61)(6 87 100 112 34 28 62)(7 88 101 105 35 29 63)(8 81 102 106 36 30 64)(9 43 80 17 51 93 72)(10 44 73 18 52 94 65)(11 45 74 19 53 95 66)(12 46 75 20 54 96 67)(13 47 76 21 55 89 68)(14 48 77 22 56 90 69)(15 41 78 23 49 91 70)(16 42 79 24 50 92 71)
(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)
(1 45 5 41)(2 44 6 48)(3 43 7 47)(4 42 8 46)(9 63 13 59)(10 62 14 58)(11 61 15 57)(12 60 16 64)(17 101 21 97)(18 100 22 104)(19 99 23 103)(20 98 24 102)(25 72 29 68)(26 71 30 67)(27 70 31 66)(28 69 32 65)(33 91 37 95)(34 90 38 94)(35 89 39 93)(36 96 40 92)(49 107 53 111)(50 106 54 110)(51 105 55 109)(52 112 56 108)(73 87 77 83)(74 86 78 82)(75 85 79 81)(76 84 80 88)

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

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

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

C7×Q16 is a maximal subgroup of   C7⋊SD32  C7⋊Q32  Q16⋊D7  Q8.D14

49 conjugacy classes

class 1  2 4A4B4C7A···7F8A8B14A···14F28A···28F28G···28R56A···56L
order124447···78814···1428···2828···2856···56
size112441···1221···12···24···42···2

49 irreducible representations

dim1111112222
type++++-
imageC1C2C2C7C14C14D4Q16C7×D4C7×Q16
kernelC7×Q16C56C7×Q8Q16C8Q8C14C7C2C1
# reps112661212612

Matrix representation of C7×Q16 in GL2(𝔽113) generated by

300
030
,
3182
3131
,
9746
4616
G:=sub<GL(2,GF(113))| [30,0,0,30],[31,31,82,31],[97,46,46,16] >;

C7×Q16 in GAP, Magma, Sage, TeX

C_7\times Q_{16}
% in TeX

G:=Group("C7xQ16");
// GroupNames label

G:=SmallGroup(112,26);
// by ID

G=gap.SmallGroup(112,26);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-2,280,301,286,1683,848,58]);
// Polycyclic

G:=Group<a,b,c|a^7=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×Q16 in TeX

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