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G = C8⋊D14  order 224 = 25·7

1st semidirect product of C8 and D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8⋊1D14, D56⋊2C2, C56⋊1C22, C4.14D28, C28.12D4, D28⋊4C22, M4(2)⋊1D7, C22.5D28, C28.32C23, Dic14⋊4C22, C4○D28⋊2C2, (C2×D28)⋊7C2, C56⋊C2⋊1C2, C7⋊1(C8⋊C22), (C2×C14).5D4, (C2×C4).15D14, C2.15(C2×D28), C14.13(C2×D4), (C7×M4(2))⋊1C2, C4.30(C22×D7), (C2×C28).27C22, SmallGroup(224,103)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C28 — C8⋊D14
Chief series
C1 — C7 — C14 — C28 — D28 — C2×D28 — C8⋊D14
Lower central
C7 — C14 — C28 — C8⋊D14
Upper central
C1 — C2 — C2×C4 — M4(2)

Generators and relations for C8⋊D14
 G = < a,b,c | a8=b14=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 398 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, M4(2), D8, SD16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C8⋊C22, C56, Dic14, C4×D7, D28, D28, D28, C7⋊D4, C2×C28, C22×D7, C56⋊C2, D56, C7×M4(2), C2×D28, C4○D28, C8⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8⋊C22, D28, C22×D7, C2×D28, C8⋊D14

Smallest permutation representation of C8⋊D14
►On 56 points
Generators in S56
(1 35 15 47 12 42 25 54)(2 29 16 55 13 36 26 48)(3 37 17 49 14 30 27 56)(4 31 18 43 8 38 28 50)(5 39 19 51 9 32 22 44)(6 33 20 45 10 40 23 52)(7 41 21 53 11 34 24 46)

(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)

(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 21)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 28)(29 39)(30 38)(31 37)(32 36)(33 35)(40 42)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)
 
G:=sub<Sym(56)| (1,35,15,47,12,42,25,54)(2,29,16,55,13,36,26,48)(3,37,17,49,14,30,27,56)(4,31,18,43,8,38,28,50)(5,39,19,51,9,32,22,44)(6,33,20,45,10,40,23,52)(7,41,21,53,11,34,24,46), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,21)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,28)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)>;
 
G:=Group( (1,35,15,47,12,42,25,54)(2,29,16,55,13,36,26,48)(3,37,17,49,14,30,27,56)(4,31,18,43,8,38,28,50)(5,39,19,51,9,32,22,44)(6,33,20,45,10,40,23,52)(7,41,21,53,11,34,24,46), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,21)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,28)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50) );
 
G=PermutationGroup([[(1,35,15,47,12,42,25,54),(2,29,16,55,13,36,26,48),(3,37,17,49,14,30,27,56),(4,31,18,43,8,38,28,50),(5,39,19,51,9,32,22,44),(6,33,20,45,10,40,23,52),(7,41,21,53,11,34,24,46)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,21),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,28),(29,39),(30,38),(31,37),(32,36),(33,35),(40,42),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50)]])
 

C8⋊D14 is a maximal subgroup of
 D28⋊1D4  D28.3D4  D28.5D4  D28.6D4  D4⋊4D28  D4.10D28  C8.21D28  C8.24D28  C56.9C23  D4.11D28  D4.12D28  D7×C8⋊C22  D8⋊5D14  D56⋊C22  C56.C23
C8⋊D14 is a maximal quotient of
 C8⋊Dic14  C42.16D14  D56⋊C4  C8⋊D28  C42.19D14  C42.20D14  C23.35D28  D28.31D4  D28⋊13D4  D28⋊14D4  C23.38D28  C23.13D28  D28⋊3Q8  C4⋊D56  D28.19D4  D28.3Q8  Dic14⋊8D4  C28.7Q16  C23.47D28  C23.48D28  C23.49D28  C56⋊2D4  C56⋊3D4

41 conjugacy classes

class 1 2A2B2C2D2E4A4B4C7A7B7C8A8B14A14B14C14D14E14F28A···28F28G28H28I56A···56L
order1222224447778814141414141428···2828282856···56
size1122828282228222442224442···24444···4

41 irreducible representations

dim111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D7D14D14D28D28C8⋊C22C8⋊D14
kernelC8⋊D14C56⋊C2D56C7×M4(2)C2×D28C4○D28C28C2×C14M4(2)C8C2×C4C4C22C7C1
# reps122111113636616

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

11204242
011262104
471110
17401
,
1032500
99100
07988
91481040
,
893400
132400
81595867
791115155
G:=sub<GL(4,GF(113))| [112,0,47,1,0,112,11,74,42,62,1,0,42,104,0,1],[103,99,0,91,25,1,7,48,0,0,9,104,0,0,88,0],[89,13,81,79,34,24,59,111,0,0,58,51,0,0,67,55] >;
 

C8⋊D14 in GAP, Magma, Sage, TeX 

C_8\rtimes D_{14}
 
% in TeX
 
G:=Group("C8:D14");
 
// GroupNames label
 
G:=SmallGroup(224,103);
 
// by ID
 
G=gap.SmallGroup(224,103);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,50,579,69,6917]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^14=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

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