Copied to
clipboard

## G = Q16×C7⋊C3order 336 = 24·3·7

### Direct product of Q16 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Q16×C7⋊C3
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — Q8×C7⋊C3 — Q16×C7⋊C3
 Lower central C7 — C14 — C28 — Q16×C7⋊C3
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16×C7⋊C3
G = < a,b,c,d | a8=c7=d3=1, b2=a4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of Q16×C7⋊C3
On 112 points
Generators in S112
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 48 13 44)(10 47 14 43)(11 46 15 42)(12 45 16 41)(17 61 21 57)(18 60 22 64)(19 59 23 63)(20 58 24 62)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(65 94 69 90)(66 93 70 89)(67 92 71 96)(68 91 72 95)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)
(1 36 61 107 89 14 102)(2 37 62 108 90 15 103)(3 38 63 109 91 16 104)(4 39 64 110 92 9 97)(5 40 57 111 93 10 98)(6 33 58 112 94 11 99)(7 34 59 105 95 12 100)(8 35 60 106 96 13 101)(17 52 70 47 78 29 84)(18 53 71 48 79 30 85)(19 54 72 41 80 31 86)(20 55 65 42 73 32 87)(21 56 66 43 74 25 88)(22 49 67 44 75 26 81)(23 50 68 45 76 27 82)(24 51 69 46 77 28 83)
(9 110 97)(10 111 98)(11 112 99)(12 105 100)(13 106 101)(14 107 102)(15 108 103)(16 109 104)(17 70 84)(18 71 85)(19 72 86)(20 65 87)(21 66 88)(22 67 81)(23 68 82)(24 69 83)(33 58 94)(34 59 95)(35 60 96)(36 61 89)(37 62 90)(38 63 91)(39 64 92)(40 57 93)(41 54 80)(42 55 73)(43 56 74)(44 49 75)(45 50 76)(46 51 77)(47 52 78)(48 53 79)

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

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

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

35 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 6A 6B 7A 7B 8A 8B 12A 12B 12C 12D 12E 12F 14A 14B 24A 24B 24C 24D 28A 28B 28C 28D 28E 28F 56A 56B 56C 56D order 1 2 3 3 4 4 4 6 6 7 7 8 8 12 12 12 12 12 12 14 14 24 24 24 24 28 28 28 28 28 28 56 56 56 56 size 1 1 7 7 2 4 4 7 7 3 3 2 2 14 14 28 28 28 28 3 3 14 14 14 14 6 6 12 12 12 12 6 6 6 6

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + - image C1 C2 C2 C3 C6 C6 D4 Q16 C3×D4 C3×Q16 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 D4×C7⋊C3 Q16×C7⋊C3 kernel Q16×C7⋊C3 C8×C7⋊C3 Q8×C7⋊C3 C7×Q16 C56 C7×Q8 C2×C7⋊C3 C7⋊C3 C14 C7 Q16 C8 Q8 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 2 2 4 2 4

Matrix representation of Q16×C7⋊C3 in GL5(𝔽337)

 0 311 0 0 0 13 311 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 333 255 0 0 0 292 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 336 212 1 0 0 0 212 1 0 0 336 213 1
,
 208 0 0 0 0 0 208 0 0 0 0 0 213 1 125 0 0 1 0 0 0 0 1 1 124

G:=sub<GL(5,GF(337))| [0,13,0,0,0,311,311,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[333,292,0,0,0,255,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,336,0,0,212,212,213,0,0,1,1,1],[208,0,0,0,0,0,208,0,0,0,0,0,213,1,1,0,0,1,0,1,0,0,125,0,124] >;

Q16×C7⋊C3 in GAP, Magma, Sage, TeX

Q_{16}\times C_7\rtimes C_3
% in TeX

G:=Group("Q16xC7:C3");
// GroupNames label

G:=SmallGroup(336,55);
// by ID

G=gap.SmallGroup(336,55);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,151,867,441,69,881]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^7=d^3=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

Export

׿
×
𝔽