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G = D7xD8order 224 = 25·7

Direct product of D7 and D8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D7xD8, C8:4D14, D56:4C2, D4:1D14, C56:2C22, D14.12D4, D28:1C22, C28.1C23, Dic7.3D4, C7:2(C2xD8), D4:D7:1C2, (D4xD7):1C2, (C8xD7):1C2, (C7xD8):2C2, C7:C8:5C22, C2.15(D4xD7), C14.27(C2xD4), (C7xD4):1C22, C4.1(C22xD7), (C4xD7).7C22, SmallGroup(224,105)

Series: Derived Chief Lower central Upper central

C1C28 — D7xD8
C1C7C14C28C4xD7D4xD7 — D7xD8
C7C14C28 — D7xD8
C1C2C4D8

Generators and relations for D7xD8
 G = < a,b,c,d | a7=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 446 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2xC4, D4, D4, C23, D7, D7, C14, C14, C2xC8, D8, D8, C2xD4, Dic7, C28, D14, D14, C2xC14, C2xD8, C7:C8, C56, C4xD7, D28, C7:D4, C7xD4, C22xD7, C8xD7, D56, D4:D7, C7xD8, D4xD7, D7xD8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2xD4, D14, C2xD8, C22xD7, D4xD7, D7xD8

Smallest permutation representation of D7xD8
On 56 points
Generators in S56
(1 26 16 53 35 24 41)(2 27 9 54 36 17 42)(3 28 10 55 37 18 43)(4 29 11 56 38 19 44)(5 30 12 49 39 20 45)(6 31 13 50 40 21 46)(7 32 14 51 33 22 47)(8 25 15 52 34 23 48)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 23)(18 22)(19 21)(25 27)(28 32)(29 31)(33 37)(34 36)(38 40)(42 48)(43 47)(44 46)(50 56)(51 55)(52 54)

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

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

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

D7xD8 is a maximal subgroup of
D8:D14  D112:C2  D8:13D14  D8:15D14  D8:5D14
D7xD8 is a maximal quotient of
Dic7:4D8  Dic7.D8  Dic7.SD16  D4:D28  D14.D8  D14:D8  D28:3D4  Dic7:5D8  C56:2Q8  D14.5D8  C8:7D28  D28:2Q8  D8:D14  D16:3D7  D112:C2  SD32:D7  SD32:3D7  Q32:D7  Q32:3D7  Dic7:D8  C56:5D4  D28:D4  C56:6D4

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B7A7B7C8A8B8C8D14A14B14C14D···14I28A28B28C56A···56F
order1222222244777888814141414···1428282856···56
size11447728282142222214142228···84444···4

35 irreducible representations

dim11111122222244
type++++++++++++++
imageC1C2C2C2C2C2D4D4D7D8D14D14D4xD7D7xD8
kernelD7xD8C8xD7D56D4:D7C7xD8D4xD7Dic7D14D8D7C8D4C2C1
# reps11121211343636

Matrix representation of D7xD8 in GL4(F113) generated by

0100
1122400
0010
0001
,
0100
1000
0010
0001
,
112000
011200
00519
00250
,
1000
0100
0010
0032112
G:=sub<GL(4,GF(113))| [0,112,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,51,25,0,0,9,0],[1,0,0,0,0,1,0,0,0,0,1,32,0,0,0,112] >;

D7xD8 in GAP, Magma, Sage, TeX

D_7\times D_8
% in TeX

G:=Group("D7xD8");
// GroupNames label

G:=SmallGroup(224,105);
// by ID

G=gap.SmallGroup(224,105);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,116,297,159,69,6917]);
// Polycyclic

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

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