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G = D14⋊C8order 224 = 25·7

The semidirect product of D14 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D14⋊C8, C4.19D28, C28.52D4, C14.3M4(2), (C2×C8)⋊1D7, (C2×C56)⋊1C2, C2.5(C8×D7), C71(C22⋊C8), C14.5(C2×C8), (C2×C4).93D14, C2.1(D14⋊C4), C2.3(C8⋊D7), C4.27(C7⋊D4), (C2×Dic7).4C4, (C22×D7).2C4, C22.11(C4×D7), C14.6(C22⋊C4), (C2×C28).107C22, (C2×C7⋊C8)⋊9C2, (C2×C4×D7).7C2, (C2×C14).12(C2×C4), SmallGroup(224,26)

Series: Derived Chief Lower central Upper central

C1C14 — D14⋊C8
C1C7C14C28C2×C28C2×C4×D7 — D14⋊C8
C7C14 — D14⋊C8
C1C2×C4C2×C8

Generators and relations for D14⋊C8
 G = < a,b,c | a14=b2=c8=1, bab=a-1, ac=ca, cbc-1=a7b >

14C2
14C2
7C22
7C22
14C22
14C22
14C4
2D7
2D7
2C8
7C2×C4
7C23
14C8
14C2×C4
14C2×C4
2D14
2Dic7
2D14
7C2×C8
7C22×C4
2C7⋊C8
2C56
2C4×D7
2C4×D7
7C22⋊C8

Smallest permutation representation of D14⋊C8
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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 100)(12 99)(13 112)(14 111)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 84)(27 83)(28 82)(29 57)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 97)(44 96)(45 95)(46 94)(47 93)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 98)
(1 38 56 20 111 70 85 84)(2 39 43 21 112 57 86 71)(3 40 44 22 99 58 87 72)(4 41 45 23 100 59 88 73)(5 42 46 24 101 60 89 74)(6 29 47 25 102 61 90 75)(7 30 48 26 103 62 91 76)(8 31 49 27 104 63 92 77)(9 32 50 28 105 64 93 78)(10 33 51 15 106 65 94 79)(11 34 52 16 107 66 95 80)(12 35 53 17 108 67 96 81)(13 36 54 18 109 68 97 82)(14 37 55 19 110 69 98 83)

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

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

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

D14⋊C8 is a maximal subgroup of
C42.282D14  C8×D28  C86D28  C42.243D14  C42.182D14  C89D28  C42.185D14  D7×C22⋊C8  C7⋊D4⋊C8  D14⋊M4(2)  D14⋊C8⋊C2  D142M4(2)  Dic7⋊M4(2)  C7⋊C826D4  D4⋊D28  D14.D8  D4.6D28  D14.SD16  C8⋊Dic7⋊C2  D43D28  D4.D28  C561C4⋊C2  D14.1SD16  Q82D28  D144Q16  D14.Q16  Q8.D28  D284D4  D14⋊C8.C2  (C2×C8).D14  C42.200D14  D28⋊C8  C42.202D14  D143M4(2)  C282M4(2)  C42.31D14  D14.2SD16  D14.4SD16  C4.Q8⋊D7  C28.(C4○D4)  D14.5D8  D14.2Q16  C2.D8⋊D7  C2.D87D7  C8×C7⋊D4  (C22×C8)⋊D7  C5632D4  D146M4(2)  C56⋊D4  C5618D4  (C2×D28).14C4  D28⋊D4  Dic14⋊D4  D146SD16  Dic147D4  D287D4  Dic14.16D4  D145Q16  D28.17D4
D14⋊C8 is a maximal quotient of
C4.8Dic28  C4.17D56  (C22×D7)⋊C8  (C2×Dic7)⋊C8  D282C8  Dic142C8  D14⋊C16  D28.C8  M5(2)⋊D7  Dic14.C8  (C2×C56)⋊5C4

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F7A7B7C8A8B8C8D8E8F8G8H14A···14I28A···28L56A···56X
order1222224444447778888888814···1428···2856···56
size11111414111114142222222141414142···22···22···2

68 irreducible representations

dim1111111222222222
type++++++++
imageC1C2C2C2C4C4C8D4D7M4(2)D14D28C7⋊D4C4×D7C8×D7C8⋊D7
kernelD14⋊C8C2×C7⋊C8C2×C56C2×C4×D7C2×Dic7C22×D7D14C28C2×C8C14C2×C4C4C4C22C2C2
# reps111122823236661212

Matrix representation of D14⋊C8 in GL4(𝔽113) generated by

348800
258800
001010
0010324
,
011200
112000
001010
0024103
,
44000
04400
0084106
00729
G:=sub<GL(4,GF(113))| [34,25,0,0,88,88,0,0,0,0,10,103,0,0,10,24],[0,112,0,0,112,0,0,0,0,0,10,24,0,0,10,103],[44,0,0,0,0,44,0,0,0,0,84,7,0,0,106,29] >;

D14⋊C8 in GAP, Magma, Sage, TeX

D_{14}\rtimes C_8
% in TeX

G:=Group("D14:C8");
// GroupNames label

G:=SmallGroup(224,26);
// by ID

G=gap.SmallGroup(224,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,86,6917]);
// Polycyclic

G:=Group<a,b,c|a^14=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations

Export

Subgroup lattice of D14⋊C8 in TeX

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