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G = C14xD8order 224 = 25·7

Direct product of C14 and D8

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

Aliases: C14xD8, C28.41D4, C56:12C22, C28.44C23, (C2xC8):3C14, C8:2(C2xC14), C4.6(C7xD4), (C2xC56):11C2, D4:1(C2xC14), (C2xD4):4C14, (D4xC14):13C2, C14.74(C2xD4), C2.11(D4xC14), (C2xC14).52D4, (C7xD4):10C22, C4.1(C22xC14), C22.14(C7xD4), (C2xC28).129C22, (C2xC4).25(C2xC14), SmallGroup(224,167)

Series: Derived Chief Lower central Upper central

C1C4 — C14xD8
C1C2C4C28C7xD4C7xD8 — C14xD8
C1C2C4 — C14xD8
C1C2xC14C2xC28 — C14xD8

Generators and relations for C14xD8
 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, C2xC4, D4, D4, C23, C14, C14, C14, C2xC8, D8, C2xD4, C28, C2xC14, C2xC14, C2xD8, C56, C2xC28, C7xD4, C7xD4, C22xC14, C2xC56, C7xD8, D4xC14, C14xD8
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2xD4, C2xC14, C2xD8, C7xD4, C22xC14, C7xD8, D4xC14, C14xD8

Smallest permutation representation of C14xD8
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)]])

C14xD8 is a maximal subgroup of
C14.SD32  D8.Dic7  D8.D14  Dic7:D8  C56:5D4  D8:Dic7  (C2xD8).D7  C56:11D4  C56.22D4  D28:D4  C56:6D4  Dic14:D4  C56:12D4  C56.23D4  D8:13D14

98 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B7A···7F8A8B8C8D14A···14R14S···14AP28A···28L56A···56X
order12222222447···7888814···1414···1428···2856···56
size11114444221···122221···14···42···22···2

98 irreducible representations

dim11111111222222
type+++++++
imageC1C2C2C2C7C14C14C14D4D4D8C7xD4C7xD4C7xD8
kernelC14xD8C2xC56C7xD8D4xC14C2xD8C2xC8D8C2xD4C28C2xC14C14C4C22C2
# reps11426624121146624

Matrix representation of C14xD8 in GL4(F113) generated by

7000
0700
00830
00083
,
011200
1000
00031
005162
,
011200
112000
006282
006251
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] >;

C14xD8 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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