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## G = D7×D8order 224 = 25·7

### Direct product of D7 and D8

Aliases: D7×D8, C84D14, D564C2, D41D14, C562C22, D14.12D4, D281C22, C28.1C23, Dic7.3D4, C72(C2×D8), D4⋊D71C2, (D4×D7)⋊1C2, (C8×D7)⋊1C2, (C7×D8)⋊2C2, C7⋊C85C22, C2.15(D4×D7), C14.27(C2×D4), (C7×D4)⋊1C22, C4.1(C22×D7), (C4×D7).7C22, SmallGroup(224,105)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×D8
 Chief series C1 — C7 — C14 — C28 — C4×D7 — D4×D7 — D7×D8
 Lower central C7 — C14 — C28 — D7×D8
 Upper central C1 — C2 — C4 — D8

Generators and relations for D7×D8
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, C2×C4, D4, D4, C23, D7, D7, C14, C14, C2×C8, D8, D8, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×D8, C7⋊C8, C56, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C8×D7, D56, D4⋊D7, C7×D8, D4×D7, D7×D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C22×D7, D4×D7, D7×D8

Smallest permutation representation of D7×D8
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)]])

D7×D8 is a maximal subgroup of
D8⋊D14  D112⋊C2  D813D14  D815D14  D85D14
D7×D8 is a maximal quotient of
Dic74D8  Dic7.D8  Dic7.SD16  D4⋊D28  D14.D8  D14⋊D8  D283D4  Dic75D8  C562Q8  D14.5D8  C87D28  D282Q8  D8⋊D14  D163D7  D112⋊C2  SD32⋊D7  SD323D7  Q32⋊D7  Q323D7  Dic7⋊D8  C565D4  D28⋊D4  C566D4

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D ··· 14I 28A 28B 28C 56A ··· 56F order 1 2 2 2 2 2 2 2 4 4 7 7 7 8 8 8 8 14 14 14 14 ··· 14 28 28 28 56 ··· 56 size 1 1 4 4 7 7 28 28 2 14 2 2 2 2 2 14 14 2 2 2 8 ··· 8 4 4 4 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D14 D14 D4×D7 D7×D8 kernel D7×D8 C8×D7 D56 D4⋊D7 C7×D8 D4×D7 Dic7 D14 D8 D7 C8 D4 C2 C1 # reps 1 1 1 2 1 2 1 1 3 4 3 6 3 6

Matrix representation of D7×D8 in GL4(𝔽113) generated by

 0 1 0 0 112 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 9 0 0 25 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 32 112
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] >;

D7×D8 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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