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## G = C8.Dic14order 448 = 26·7

### 2nd non-split extension by C8 of Dic14 acting via Dic14/C14=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C8.Dic14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C28.C8 — C8.Dic14
 Lower central C7 — C14 — C28 — C56 — C8.Dic14
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for C8.Dic14
G = < a,b,c | a8=1, b28=a4, c2=ab14, bab-1=a-1, cac-1=a5, cbc-1=a-1b27 >

Smallest permutation representation of C8.Dic14
On 112 points
Generators in S112
(1 110 43 96 29 82 15 68)(2 69 16 83 30 97 44 111)(3 112 45 98 31 84 17 70)(4 71 18 85 32 99 46 57)(5 58 47 100 33 86 19 72)(6 73 20 87 34 101 48 59)(7 60 49 102 35 88 21 74)(8 75 22 89 36 103 50 61)(9 62 51 104 37 90 23 76)(10 77 24 91 38 105 52 63)(11 64 53 106 39 92 25 78)(12 79 26 93 40 107 54 65)(13 66 55 108 41 94 27 80)(14 81 28 95 42 109 56 67)
(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 22 68 103 15 8 82 89 29 50 96 75 43 36 110 61)(2 88 83 35 44 102 69 49 30 60 111 7 16 74 97 21)(3 20 70 101 17 6 84 87 31 48 98 73 45 34 112 59)(4 86 85 33 46 100 71 47 32 58 57 5 18 72 99 19)(9 14 76 95 23 56 90 81 37 42 104 67 51 28 62 109)(10 80 91 27 52 94 77 41 38 108 63 55 24 66 105 13)(11 12 78 93 25 54 92 79 39 40 106 65 53 26 64 107)

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

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

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

58 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 ··· 56X 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 size 1 1 2 2 2 56 56 2 2 2 2 2 4 8 8 2 2 2 4 4 4 28 28 28 28 2 ··· 2 4 4 4 4 ··· 4 8 ··· 8

58 irreducible representations

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

Matrix representation of C8.Dic14 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 26 16 0 0 0 0 106 0 0 0 0 0 70 17 100 13 0 0 27 17 100 100
,
 0 98 0 0 0 0 15 58 0 0 0 0 0 0 103 0 36 0 0 0 0 0 112 1 0 0 38 112 10 0 0 0 82 0 10 0
,
 112 0 0 0 0 0 34 1 0 0 0 0 0 0 0 0 97 16 0 0 43 0 26 0 0 0 91 112 96 17 0 0 91 0 96 17

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,26,106,70,27,0,0,16,0,17,17,0,0,0,0,100,100,0,0,0,0,13,100],[0,15,0,0,0,0,98,58,0,0,0,0,0,0,103,0,38,82,0,0,0,0,112,0,0,0,36,112,10,10,0,0,0,1,0,0],[112,34,0,0,0,0,0,1,0,0,0,0,0,0,0,43,91,91,0,0,0,0,112,0,0,0,97,26,96,96,0,0,16,0,17,17] >;

C8.Dic14 in GAP, Magma, Sage, TeX

C_8.{\rm Dic}_{14}
% in TeX

G:=Group("C8.Dic14");
// GroupNames label

G:=SmallGroup(448,51);
// by ID

G=gap.SmallGroup(448,51);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,365,36,758,184,346,80,851,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^28=a^4,c^2=a*b^14,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^-1*b^27>;
// generators/relations

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