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## G = C14×D8order 224 = 25·7

### Direct product of C14 and D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C14×D8
 Chief series C1 — C2 — C4 — C28 — C7×D4 — C7×D8 — C14×D8
 Lower central C1 — C2 — C4 — C14×D8
 Upper central C1 — C2×C14 — C2×C28 — C14×D8

Generators and relations for C14×D8
G = < a,b,c | a14=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, D4, C23, C14, C14, C14, C2×C8, D8, C2×D4, C28, C2×C14, C2×C14, C2×D8, C56, C2×C28, C7×D4, C7×D4, C22×C14, C2×C56, C7×D8, D4×C14, C14×D8
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, C2×D8, C7×D4, C22×C14, C7×D8, D4×C14, C14×D8

Smallest permutation representation of C14×D8
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 82 66 102 47 18 33 93)(2 83 67 103 48 19 34 94)(3 84 68 104 49 20 35 95)(4 71 69 105 50 21 36 96)(5 72 70 106 51 22 37 97)(6 73 57 107 52 23 38 98)(7 74 58 108 53 24 39 85)(8 75 59 109 54 25 40 86)(9 76 60 110 55 26 41 87)(10 77 61 111 56 27 42 88)(11 78 62 112 43 28 29 89)(12 79 63 99 44 15 30 90)(13 80 64 100 45 16 31 91)(14 81 65 101 46 17 32 92)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 112)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)

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

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

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

C14×D8 is a maximal subgroup of
C14.SD32  D8.Dic7  D8.D14  Dic7⋊D8  C565D4  D8⋊Dic7  (C2×D8).D7  C5611D4  C56.22D4  D28⋊D4  C566D4  Dic14⋊D4  C5612D4  C56.23D4  D813D14

98 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AP 28A ··· 28L 56A ··· 56X order 1 2 2 2 2 2 2 2 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 4 4 2 2 1 ··· 1 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 D4 D8 C7×D4 C7×D4 C7×D8 kernel C14×D8 C2×C56 C7×D8 D4×C14 C2×D8 C2×C8 D8 C2×D4 C28 C2×C14 C14 C4 C22 C2 # reps 1 1 4 2 6 6 24 12 1 1 4 6 6 24

Matrix representation of C14×D8 in GL4(𝔽113) generated by

 7 0 0 0 0 7 0 0 0 0 83 0 0 0 0 83
,
 0 112 0 0 1 0 0 0 0 0 0 31 0 0 51 62
,
 0 112 0 0 112 0 0 0 0 0 62 82 0 0 62 51
G:=sub<GL(4,GF(113))| [7,0,0,0,0,7,0,0,0,0,83,0,0,0,0,83],[0,1,0,0,112,0,0,0,0,0,0,51,0,0,31,62],[0,112,0,0,112,0,0,0,0,0,62,62,0,0,82,51] >;

C14×D8 in GAP, Magma, Sage, TeX

C_{14}\times D_8
% in TeX

G:=Group("C14xD8");
// GroupNames label

G:=SmallGroup(224,167);
// by ID

G=gap.SmallGroup(224,167);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,5044,2530,88]);
// Polycyclic

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

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