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G = C8×F7order 336 = 24·3·7

Direct product of C8 and F7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C8×F7
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C4×F7 — C8×F7
 Lower central C7 — C8×F7
 Upper central C1 — C8

Generators and relations for C8×F7
G = < a,b,c | a8=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C8×F7
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 31 13 37 48 17 49)(2 32 14 38 41 18 50)(3 25 15 39 42 19 51)(4 26 16 40 43 20 52)(5 27 9 33 44 21 53)(6 28 10 34 45 22 54)(7 29 11 35 46 23 55)(8 30 12 36 47 24 56)
(9 53 44 21 27 33)(10 54 45 22 28 34)(11 55 46 23 29 35)(12 56 47 24 30 36)(13 49 48 17 31 37)(14 50 41 18 32 38)(15 51 42 19 25 39)(16 52 43 20 26 40)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A ··· 6F 7 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 14 24A ··· 24P 28A 28B 56A 56B 56C 56D order 1 2 2 2 3 3 4 4 4 4 6 ··· 6 7 8 8 8 8 8 8 8 8 12 ··· 12 14 24 ··· 24 28 28 56 56 56 56 size 1 1 7 7 7 7 1 1 7 7 7 ··· 7 6 1 1 1 1 7 7 7 7 7 ··· 7 6 7 ··· 7 6 6 6 6 6 6

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 F7 C2×F7 C4×F7 C8×F7 kernel C8×F7 C7⋊C24 C8×C7⋊C3 C4×F7 C8×D7 C7⋊C12 C2×F7 C7⋊C8 C56 C4×D7 F7 Dic7 D14 D7 C8 C4 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 8 4 4 16 1 1 2 4

Matrix representation of C8×F7 in GL6(𝔽337)

 85 0 0 0 0 0 0 85 0 0 0 0 0 0 85 0 0 0 0 0 0 85 0 0 0 0 0 0 85 0 0 0 0 0 0 85
,
 336 336 336 336 336 336 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0

G:=sub<GL(6,GF(337))| [85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C8×F7 in GAP, Magma, Sage, TeX

C_8\times F_7
% in TeX

G:=Group("C8xF7");
// GroupNames label

G:=SmallGroup(336,7);
// by ID

G=gap.SmallGroup(336,7);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,79,69,10373,1745]);
// Polycyclic

G:=Group<a,b,c|a^8=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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