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## G = C8×D8order 128 = 27

### Direct product of C8 and D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C8×D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8×D4 — C8×D8
 Lower central C1 — C2 — C4 — C8×D8
 Upper central C1 — C2×C8 — C4×C8 — C8×D8
 Jennings C1 — C22 — C22 — C42 — C8×D8

Generators and relations for C8×D8
G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 184 in 98 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C22×C8, C2×D8, C82, D4⋊C8, C81C8, C8×D4, C4×D8, C8×D8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, D8, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C2×D8, C4○D8, C8×D4, C4×D8, C8○D8, C8×D8

Smallest permutation representation of C8×D8
On 64 points
Generators in S64
(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)(57 58 59 60 61 62 63 64)
(1 55 16 21 31 63 39 48)(2 56 9 22 32 64 40 41)(3 49 10 23 25 57 33 42)(4 50 11 24 26 58 34 43)(5 51 12 17 27 59 35 44)(6 52 13 18 28 60 36 45)(7 53 14 19 29 61 37 46)(8 54 15 20 30 62 38 47)
(1 5)(2 6)(3 7)(4 8)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 29)(26 30)(27 31)(28 32)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)

G:=sub<Sym(64)| (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)(57,58,59,60,61,62,63,64), (1,55,16,21,31,63,39,48)(2,56,9,22,32,64,40,41)(3,49,10,23,25,57,33,42)(4,50,11,24,26,58,34,43)(5,51,12,17,27,59,35,44)(6,52,13,18,28,60,36,45)(7,53,14,19,29,61,37,46)(8,54,15,20,30,62,38,47), (1,5)(2,6)(3,7)(4,8)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,29)(26,30)(27,31)(28,32)(41,52)(42,53)(43,54)(44,55)(45,56)(46,49)(47,50)(48,51)>;

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)(57,58,59,60,61,62,63,64), (1,55,16,21,31,63,39,48)(2,56,9,22,32,64,40,41)(3,49,10,23,25,57,33,42)(4,50,11,24,26,58,34,43)(5,51,12,17,27,59,35,44)(6,52,13,18,28,60,36,45)(7,53,14,19,29,61,37,46)(8,54,15,20,30,62,38,47), (1,5)(2,6)(3,7)(4,8)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,29)(26,30)(27,31)(28,32)(41,52)(42,53)(43,54)(44,55)(45,56)(46,49)(47,50)(48,51) );

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),(57,58,59,60,61,62,63,64)], [(1,55,16,21,31,63,39,48),(2,56,9,22,32,64,40,41),(3,49,10,23,25,57,33,42),(4,50,11,24,26,58,34,43),(5,51,12,17,27,59,35,44),(6,52,13,18,28,60,36,45),(7,53,14,19,29,61,37,46),(8,54,15,20,30,62,38,47)], [(1,5),(2,6),(3,7),(4,8),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,29),(26,30),(27,31),(28,32),(41,52),(42,53),(43,54),(44,55),(45,56),(46,49),(47,50),(48,51)]])

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A ··· 8H 8I ··· 8AB 8AC ··· 8AJ order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 ··· 8 8 ··· 8 size 1 1 1 1 4 4 4 4 1 1 1 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 D4 D8 C4○D4 C8○D4 C4○D8 C8○D8 kernel C8×D8 C82 D4⋊C8 C8⋊1C8 C8×D4 C4×D8 D4⋊C4 C2.D8 C2×D8 D8 C2×C8 C8 C2×C4 C4 C4 C2 # reps 1 1 2 1 2 1 4 2 2 16 2 4 2 4 4 8

Matrix representation of C8×D8 in GL4(𝔽17) generated by

 15 0 0 0 0 15 0 0 0 0 13 0 0 0 0 13
,
 9 16 0 0 14 8 0 0 0 0 9 0 0 0 11 2
,
 16 0 0 0 16 1 0 0 0 0 1 4 0 0 0 16
G:=sub<GL(4,GF(17))| [15,0,0,0,0,15,0,0,0,0,13,0,0,0,0,13],[9,14,0,0,16,8,0,0,0,0,9,11,0,0,0,2],[16,16,0,0,0,1,0,0,0,0,1,0,0,0,4,16] >;

C8×D8 in GAP, Magma, Sage, TeX

C_8\times D_8
% in TeX

G:=Group("C8xD8");
// GroupNames label

G:=SmallGroup(128,307);
// by ID

G=gap.SmallGroup(128,307);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,100,1123,570,136,172]);
// Polycyclic

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

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