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## G = C14×C4.D4order 448 = 26·7

### Direct product of C14 and C4.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C14×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4.D4 — C14×C4.D4
 Lower central C1 — C2 — C22 — C14×C4.D4
 Upper central C1 — C2×C14 — C22×C28 — C14×C4.D4

Generators and relations for C14×C4.D4
G = < a,b,c,d | a14=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 370 in 186 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, C28, C2×C14, C2×C14, C4.D4, C2×M4(2), C22×D4, C56, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C4.D4, C2×C56, C7×M4(2), C7×M4(2), C22×C28, D4×C14, D4×C14, C23×C14, C7×C4.D4, C14×M4(2), D4×C2×C14, C14×C4.D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C4.D4, C2×C22⋊C4, C2×C28, C7×D4, C22×C14, C2×C4.D4, C7×C22⋊C4, C22×C28, D4×C14, C7×C4.D4, C14×C22⋊C4, C14×C4.D4

Smallest permutation representation of C14×C4.D4
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 40 78 86)(2 41 79 87)(3 42 80 88)(4 29 81 89)(5 30 82 90)(6 31 83 91)(7 32 84 92)(8 33 71 93)(9 34 72 94)(10 35 73 95)(11 36 74 96)(12 37 75 97)(13 38 76 98)(14 39 77 85)(15 63 44 107)(16 64 45 108)(17 65 46 109)(18 66 47 110)(19 67 48 111)(20 68 49 112)(21 69 50 99)(22 70 51 100)(23 57 52 101)(24 58 53 102)(25 59 54 103)(26 60 55 104)(27 61 56 105)(28 62 43 106)
(1 53 93 109 78 24 33 65)(2 54 94 110 79 25 34 66)(3 55 95 111 80 26 35 67)(4 56 96 112 81 27 36 68)(5 43 97 99 82 28 37 69)(6 44 98 100 83 15 38 70)(7 45 85 101 84 16 39 57)(8 46 86 102 71 17 40 58)(9 47 87 103 72 18 41 59)(10 48 88 104 73 19 42 60)(11 49 89 105 74 20 29 61)(12 50 90 106 75 21 30 62)(13 51 91 107 76 22 31 63)(14 52 92 108 77 23 32 64)
(1 24 40 58 78 53 86 102)(2 25 41 59 79 54 87 103)(3 26 42 60 80 55 88 104)(4 27 29 61 81 56 89 105)(5 28 30 62 82 43 90 106)(6 15 31 63 83 44 91 107)(7 16 32 64 84 45 92 108)(8 17 33 65 71 46 93 109)(9 18 34 66 72 47 94 110)(10 19 35 67 73 48 95 111)(11 20 36 68 74 49 96 112)(12 21 37 69 75 50 97 99)(13 22 38 70 76 51 98 100)(14 23 39 57 77 52 85 101)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 7A ··· 7F 8A ··· 8H 14A ··· 14R 14S ··· 14AD 14AE ··· 14BB 28A ··· 28X 56A ··· 56AV order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 D4 C7×D4 C4.D4 C7×C4.D4 kernel C14×C4.D4 C7×C4.D4 C14×M4(2) D4×C2×C14 D4×C14 C23×C14 C2×C4.D4 C4.D4 C2×M4(2) C22×D4 C2×D4 C24 C2×C28 C2×C4 C14 C2 # reps 1 4 2 1 4 4 6 24 12 6 24 24 4 24 2 12

Matrix representation of C14×C4.D4 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 85 0 0 0 0 0 0 85 0 0 0 0 0 0 85 0 0 0 0 0 0 85
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 112 0
,
 64 111 0 0 0 0 70 49 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 1 0 0 0 0 1 0 0 0
,
 49 2 0 0 0 0 42 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 112 0 0 0

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[64,70,0,0,0,0,111,49,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,112,0,0],[49,42,0,0,0,0,2,64,0,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C14×C4.D4 in GAP, Magma, Sage, TeX

C_{14}\times C_4.D_4
% in TeX

G:=Group("C14xC4.D4");
// GroupNames label

G:=SmallGroup(448,819);
// by ID

G=gap.SmallGroup(448,819);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,9804,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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