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## G = C6×F8order 336 = 24·3·7

### Direct product of C6 and F8

Aliases: C6×F8, C24⋊C21, C23⋊C42, (C23×C6)⋊C7, (C22×C6)⋊2C14, SmallGroup(336,213)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C6×F8
 Chief series C1 — C23 — F8 — C3×F8 — C6×F8
 Lower central C23 — C6×F8
 Upper central C1 — C6

Generators and relations for C6×F8
G = < a,b,c,d,e | a6=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 C6×F8
On 42 points
Generators in S42
(1 33 38 19 24 12)(2 34 39 20 25 13)(3 35 40 21 26 14)(4 29 41 15 27 8)(5 30 42 16 28 9)(6 31 36 17 22 10)(7 32 37 18 23 11)
(2 20)(4 15)(5 16)(6 17)(8 41)(9 42)(10 36)(13 39)(22 31)(25 34)(27 29)(28 30)
(3 21)(5 16)(6 17)(7 18)(9 42)(10 36)(11 37)(14 40)(22 31)(23 32)(26 35)(28 30)
(1 19)(4 15)(6 17)(7 18)(8 41)(10 36)(11 37)(12 38)(22 31)(23 32)(24 33)(27 29)
(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)

G:=sub<Sym(42)| (1,33,38,19,24,12)(2,34,39,20,25,13)(3,35,40,21,26,14)(4,29,41,15,27,8)(5,30,42,16,28,9)(6,31,36,17,22,10)(7,32,37,18,23,11), (2,20)(4,15)(5,16)(6,17)(8,41)(9,42)(10,36)(13,39)(22,31)(25,34)(27,29)(28,30), (3,21)(5,16)(6,17)(7,18)(9,42)(10,36)(11,37)(14,40)(22,31)(23,32)(26,35)(28,30), (1,19)(4,15)(6,17)(7,18)(8,41)(10,36)(11,37)(12,38)(22,31)(23,32)(24,33)(27,29), (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)>;

G:=Group( (1,33,38,19,24,12)(2,34,39,20,25,13)(3,35,40,21,26,14)(4,29,41,15,27,8)(5,30,42,16,28,9)(6,31,36,17,22,10)(7,32,37,18,23,11), (2,20)(4,15)(5,16)(6,17)(8,41)(9,42)(10,36)(13,39)(22,31)(25,34)(27,29)(28,30), (3,21)(5,16)(6,17)(7,18)(9,42)(10,36)(11,37)(14,40)(22,31)(23,32)(26,35)(28,30), (1,19)(4,15)(6,17)(7,18)(8,41)(10,36)(11,37)(12,38)(22,31)(23,32)(24,33)(27,29), (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) );

G=PermutationGroup([[(1,33,38,19,24,12),(2,34,39,20,25,13),(3,35,40,21,26,14),(4,29,41,15,27,8),(5,30,42,16,28,9),(6,31,36,17,22,10),(7,32,37,18,23,11)], [(2,20),(4,15),(5,16),(6,17),(8,41),(9,42),(10,36),(13,39),(22,31),(25,34),(27,29),(28,30)], [(3,21),(5,16),(6,17),(7,18),(9,42),(10,36),(11,37),(14,40),(22,31),(23,32),(26,35),(28,30)], [(1,19),(4,15),(6,17),(7,18),(8,41),(10,36),(11,37),(12,38),(22,31),(23,32),(24,33),(27,29)], [(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)]])

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 7A ··· 7F 14A ··· 14F 21A ··· 21L 42A ··· 42L order 1 2 2 2 3 3 6 6 6 6 6 6 7 ··· 7 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 7 7 1 1 1 1 7 7 7 7 8 ··· 8 8 ··· 8 8 ··· 8 8 ··· 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 7 7 7 7 type + + + + image C1 C2 C3 C6 C7 C14 C21 C42 F8 C2×F8 C3×F8 C6×F8 kernel C6×F8 C3×F8 C2×F8 F8 C23×C6 C22×C6 C24 C23 C6 C3 C2 C1 # reps 1 1 2 2 6 6 12 12 1 1 2 2

Matrix representation of C6×F8 in GL7(𝔽43)

 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 16 0 42 0 0 0 0 21 0 0 42 0 0 0 41 0 0 0 42 0 0 0 0 0 0 0 1 0 11 0 0 0 0 0 42
,
 1 0 0 0 0 0 0 4 42 0 0 0 0 0 16 0 42 0 0 0 0 21 0 0 42 0 0 0 0 0 0 0 1 0 0 35 0 0 0 0 42 0 0 0 0 0 0 0 1
,
 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 22 0 0 1 0 0 0 0 0 0 0 42 0 0 8 0 0 0 0 1 0 32 0 0 0 0 0 1
,
 4 41 0 0 0 0 0 0 39 1 0 0 0 0 0 27 0 1 0 0 0 0 22 0 0 1 0 0 0 2 0 0 0 1 0 0 8 0 0 0 0 1 0 32 0 0 0 0 0

G:=sub<GL(7,GF(43))| [7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7],[1,0,16,21,41,0,11,0,1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42],[1,4,16,21,0,35,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1],[42,0,0,22,0,8,32,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,41,39,27,22,2,8,32,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C6×F8 in GAP, Magma, Sage, TeX

C_6\times F_8
% in TeX

G:=Group("C6xF8");
// GroupNames label

G:=SmallGroup(336,213);
// by ID

G=gap.SmallGroup(336,213);
# by ID

G:=PCGroup([6,-2,-3,-7,-2,2,2,351,856,1277]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=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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