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G = C8⋊D14order 224 = 25·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C81D14, D562C2, C561C22, C4.14D28, C28.12D4, D284C22, M4(2)⋊1D7, C22.5D28, C28.32C23, Dic144C22, C4○D282C2, (C2×D28)⋊7C2, C56⋊C21C2, C71(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

C1C28 — C8⋊D14
C1C7C14C28D28C2×D28 — C8⋊D14
C7C14C28 — C8⋊D14
C1C2C2×C4M4(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 [×4], C4 [×2], C4, C22, C22 [×5], C7, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D7 [×3], C14, C14, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic7, C28 [×2], D14 [×5], C2×C14, C8⋊C22, C56 [×2], Dic14, C4×D7, D28, D28 [×2], D28, C7⋊D4, C2×C28, C22×D7, C56⋊C2 [×2], D56 [×2], C7×M4(2), C2×D28, C4○D28, C8⋊D14
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C8⋊C22, D28 [×2], C22×D7, C2×D28, C8⋊D14

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

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

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

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

C8⋊D14 is a maximal subgroup of
D281D4  D28.3D4  D28.5D4  D28.6D4  D44D28  D4.10D28  C8.21D28  C8.24D28  C56.9C23  D4.11D28  D4.12D28  D7×C8⋊C22  D85D14  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  D2813D4  D2814D4  C23.38D28  C23.13D28  D283Q8  C4⋊D56  D28.19D4  D28.3Q8  Dic148D4  C28.7Q16  C23.47D28  C23.48D28  C23.49D28  C562D4  C563D4

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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