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## G = C8×C7⋊C3order 168 = 23·3·7

### Direct product of C8 and C7⋊C3

Aliases: C8×C7⋊C3, C56⋊C3, C72C24, C28.4C6, C14.2C12, C2.(C4×C7⋊C3), C4.2(C2×C7⋊C3), (C4×C7⋊C3).4C2, (C2×C7⋊C3).2C4, SmallGroup(168,2)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C8×C7⋊C3
G = < a,b,c | a8=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C8×C7⋊C3
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 34 16 19 46 49)(2 32 35 9 20 47 50)(3 25 36 10 21 48 51)(4 26 37 11 22 41 52)(5 27 38 12 23 42 53)(6 28 39 13 24 43 54)(7 29 40 14 17 44 55)(8 30 33 15 18 45 56)
(9 50 47)(10 51 48)(11 52 41)(12 53 42)(13 54 43)(14 55 44)(15 56 45)(16 49 46)(17 29 40)(18 30 33)(19 31 34)(20 32 35)(21 25 36)(22 26 37)(23 27 38)(24 28 39)

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

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

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

C8×C7⋊C3 is a maximal subgroup of   C7⋊C48  C8⋊F7  C56⋊C6  D56⋊C3  C8.F7

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C4 C6 C8 C12 C24 C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 C8×C7⋊C3 kernel C8×C7⋊C3 C4×C7⋊C3 C56 C2×C7⋊C3 C28 C7⋊C3 C14 C7 C8 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 2 2 4 8

Matrix representation of C8×C7⋊C3 in GL3(𝔽337) generated by

 85 0 0 0 85 0 0 0 85
,
 212 213 1 1 0 0 0 1 0
,
 1 0 0 124 336 336 0 1 0
G:=sub<GL(3,GF(337))| [85,0,0,0,85,0,0,0,85],[212,1,0,213,0,1,1,0,0],[1,124,0,0,336,1,0,336,0] >;

C8×C7⋊C3 in GAP, Magma, Sage, TeX

C_8\times C_7\rtimes C_3
% in TeX

G:=Group("C8xC7:C3");
// GroupNames label

G:=SmallGroup(168,2);
// by ID

G=gap.SmallGroup(168,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-7,30,42,609]);
// Polycyclic

G:=Group<a,b,c|a^8=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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