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## G = C16⋊Dic7order 448 = 26·7

### 1st semidirect product of C16 and Dic7 acting via Dic7/C7=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C16⋊Dic7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C8⋊Dic7 — C16⋊Dic7
 Lower central C7 — C14 — C28 — C56 — C16⋊Dic7
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for C16⋊Dic7
G = < a,b,c | a16=b14=1, c2=b7, bab-1=a9, cac-1=a11, cbc-1=b-1 >

Smallest permutation representation of C16⋊Dic7
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 93 78 49 100 21 47)(2 86 79 58 101 30 48 10 94 71 50 109 22 40)(3 95 80 51 102 23 33)(4 88 65 60 103 32 34 12 96 73 52 111 24 42)(5 81 66 53 104 25 35)(6 90 67 62 105 18 36 14 82 75 54 97 26 44)(7 83 68 55 106 27 37)(8 92 69 64 107 20 38 16 84 77 56 99 28 46)(9 85 70 57 108 29 39)(11 87 72 59 110 31 41)(13 89 74 61 112 17 43)(15 91 76 63 98 19 45)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 66)(18 69 26 77)(19 72)(20 75 28 67)(21 78)(22 65 30 73)(23 68)(24 71 32 79)(25 74)(27 80)(29 70)(31 76)(33 83)(34 86 42 94)(35 89)(36 92 44 84)(37 95)(38 82 46 90)(39 85)(40 88 48 96)(41 91)(43 81)(45 87)(47 93)(49 100)(50 103 58 111)(51 106)(52 109 60 101)(53 112)(54 99 62 107)(55 102)(56 105 64 97)(57 108)(59 98)(61 104)(63 110)

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

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

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

76 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 8E 14A 14B 14C 14D 14E 14F 16A 16B 16C 16D 28A ··· 28F 28G 28H 28I 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 4 4 4 4 7 7 7 8 8 8 8 8 14 14 14 14 14 14 16 16 16 16 28 ··· 28 28 28 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 2 2 56 56 2 2 2 2 2 4 56 56 2 2 2 4 4 4 4 4 4 4 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

76 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + + - + - + image C1 C2 C2 C2 C4 Q8 D4 D7 SD16 SD16 Dic7 D14 Dic14 D28 C56⋊C2 C56⋊C2 C8.Q8 C16⋊Dic7 kernel C16⋊Dic7 C8⋊Dic7 C56.C4 C7×M5(2) C112 C56 C2×C28 M5(2) C28 C2×C14 C16 C2×C8 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 1 1 4 1 1 3 2 2 6 3 6 6 12 12 2 12

Matrix representation of C16⋊Dic7 in GL4(𝔽113) generated by

 50 42 111 0 71 89 0 111 4 73 63 71 40 96 42 24
,
 0 1 0 0 112 9 0 0 71 89 0 112 24 81 1 104
,
 1 0 0 0 9 112 0 0 17 73 63 71 0 96 73 50
G:=sub<GL(4,GF(113))| [50,71,4,40,42,89,73,96,111,0,63,42,0,111,71,24],[0,112,71,24,1,9,89,81,0,0,0,1,0,0,112,104],[1,9,17,0,0,112,73,96,0,0,63,73,0,0,71,50] >;

C16⋊Dic7 in GAP, Magma, Sage, TeX

C_{16}\rtimes {\rm Dic}_7
% in TeX

G:=Group("C16:Dic7");
// GroupNames label

G:=SmallGroup(448,70);
// by ID

G=gap.SmallGroup(448,70);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,64,387,675,80,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^16=b^14=1,c^2=b^7,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=b^-1>;
// generators/relations

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