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## G = D8×C7⋊C3order 336 = 24·3·7

### Direct product of D8 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D8×C7⋊C3
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — D4×C7⋊C3 — D8×C7⋊C3
 Lower central C7 — C14 — C28 — D8×C7⋊C3
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8×C7⋊C3
G = < a,b,c,d | a8=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of D8×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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 22)(18 21)(19 20)(23 24)(25 28)(26 27)(29 32)(30 31)(33 40)(34 39)(35 38)(36 37)(41 46)(42 45)(43 44)(47 48)(49 56)(50 55)(51 54)(52 53)
(1 27 24 44 33 10 49)(2 28 17 45 34 11 50)(3 29 18 46 35 12 51)(4 30 19 47 36 13 52)(5 31 20 48 37 14 53)(6 32 21 41 38 15 54)(7 25 22 42 39 16 55)(8 26 23 43 40 9 56)
(9 43 56)(10 44 49)(11 45 50)(12 46 51)(13 47 52)(14 48 53)(15 41 54)(16 42 55)(17 34 28)(18 35 29)(19 36 30)(20 37 31)(21 38 32)(22 39 25)(23 40 26)(24 33 27)

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

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

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

35 conjugacy classes

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

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 C3×D8 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 D4×C7⋊C3 D8×C7⋊C3 kernel D8×C7⋊C3 C8×C7⋊C3 D4×C7⋊C3 C7×D8 C56 C7×D4 C2×C7⋊C3 C7⋊C3 C14 C7 D8 C8 D4 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 2 2 4 2 4

Matrix representation of D8×C7⋊C3 in GL5(𝔽337)

 324 13 0 0 0 324 324 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 324 13 0 0 0 13 13 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 125 0 0 0 1 124
,
 128 0 0 0 0 0 128 0 0 0 0 0 124 212 0 0 0 336 213 1 0 0 336 1 0

G:=sub<GL(5,GF(337))| [324,324,0,0,0,13,324,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[324,13,0,0,0,13,13,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,125,124],[128,0,0,0,0,0,128,0,0,0,0,0,124,336,336,0,0,212,213,1,0,0,0,1,0] >;

D8×C7⋊C3 in GAP, Magma, Sage, TeX

D_8\times C_7\rtimes C_3
% in TeX

G:=Group("D8xC7:C3");
// GroupNames label

G:=SmallGroup(336,53);
// by ID

G=gap.SmallGroup(336,53);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,867,441,69,881]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^7=d^3=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^-1=c^4>;
// generators/relations

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