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## G = C8×F8order 448 = 26·7

### Direct product of C8 and F8

Aliases: C8×F8, C23⋊C56, C24.C28, (C23×C8)⋊C7, C2.(C4×F8), (C2×F8).C4, C4.2(C2×F8), (C4×F8).2C2, (C23×C4).2C14, SmallGroup(448,919)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C8×F8
 Chief series C1 — C23 — C24 — C23×C4 — C4×F8 — C8×F8
 Lower central C23 — C8×F8
 Upper central C1 — C8

Generators and relations for C8×F8
G = < a,b,c,d,e | a8=b2=c2=d2=e7=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=dc=cd, ece-1=b, ede-1=c >

Smallest permutation representation of C8×F8
On 56 points
Generators in S56
(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 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 5)(2 6)(3 7)(4 8)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)
(1 31 38 55 23 47 11)(2 32 39 56 24 48 12)(3 25 40 49 17 41 13)(4 26 33 50 18 42 14)(5 27 34 51 19 43 15)(6 28 35 52 20 44 16)(7 29 36 53 21 45 9)(8 30 37 54 22 46 10)

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 7 7 7 7 type + + + + image C1 C2 C4 C7 C8 C14 C28 C56 F8 C2×F8 C4×F8 C8×F8 kernel C8×F8 C4×F8 C2×F8 C23×C8 F8 C23×C4 C24 C23 C8 C4 C2 C1 # reps 1 1 2 6 4 6 12 24 1 1 2 4

Matrix representation of C8×F8 in GL7(𝔽113)

 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18
,
 112 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 112 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112
,
 112 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 112
,
 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0

G:=sub<GL(7,GF(113))| [18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18],[112,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112],[0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C8×F8 in GAP, Magma, Sage, TeX

C_8\times F_8
% in TeX

G:=Group("C8xF8");
// GroupNames label

G:=SmallGroup(448,919);
// by ID

G=gap.SmallGroup(448,919);
# by ID

G:=PCGroup([7,-2,-7,-2,-2,-2,2,2,98,58,1971,4716,6873]);
// Polycyclic

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

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