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## G = C2×C8.17D4order 128 = 27

### Direct product of C2 and C8.17D4

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×C8.17D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C22×Q16 — C2×C8.17D4
 Lower central C1 — C2 — C4 — C8 — C2×C8.17D4
 Upper central C1 — C22 — C22×C4 — C22×C8 — C2×C8.17D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C8.17D4

Generators and relations for C2×C8.17D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 212 in 108 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×6], Q8 [×10], C23, C16 [×2], C2×C8 [×6], C2×C8, M4(2) [×3], Q16 [×4], Q16 [×6], C22×C4, C22×C4, C2×Q8 [×9], C8.C4 [×2], C8.C4, C2×C16, M5(2) [×2], M5(2), C22×C8, C2×M4(2), C2×Q16 [×6], C2×Q16 [×3], C22×Q8, C8.17D4 [×4], C2×C8.C4, C2×M5(2), C22×Q16, C2×C8.17D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C8.17D4 [×2], C2×D4⋊C4, C2×C8.17D4

Smallest permutation representation of C2×C8.17D4
On 64 points
Generators in S64
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 49)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)
(1 19 13 23 9 27 5 31)(2 22 6 18 10 30 14 26)(3 17 15 21 11 25 7 29)(4 20 8 32 12 28 16 24)(33 58 37 54 41 50 45 62)(34 53 46 57 42 61 38 49)(35 56 39 52 43 64 47 60)(36 51 48 55 44 59 40 63)
(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)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A ··· 16H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 2 2 2 2 8 8 8 8 2 2 2 2 4 4 8 8 8 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C4 D4 D4 D8 SD16 SD16 C8.17D4 kernel C2×C8.17D4 C8.17D4 C2×C8.C4 C2×M5(2) C22×Q16 C2×Q16 C2×C8 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 8 3 1 4 2 2 4

Matrix representation of C2×C8.17D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 14 3 0 0 0 0 14 14 0 0 0 0 7 5 14 14 0 0 12 7 3 14
,
 8 8 0 0 0 0 11 9 0 0 0 0 0 0 6 12 2 0 0 0 13 2 0 2 0 0 6 11 11 5 0 0 8 5 4 15
,
 9 9 0 0 0 0 10 8 0 0 0 0 0 0 7 14 8 0 0 0 16 9 0 9 0 0 8 12 10 14 0 0 4 13 16 8

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,14,7,12,0,0,3,14,5,7,0,0,0,0,14,3,0,0,0,0,14,14],[8,11,0,0,0,0,8,9,0,0,0,0,0,0,6,13,6,8,0,0,12,2,11,5,0,0,2,0,11,4,0,0,0,2,5,15],[9,10,0,0,0,0,9,8,0,0,0,0,0,0,7,16,8,4,0,0,14,9,12,13,0,0,8,0,10,16,0,0,0,9,14,8] >;

C2×C8.17D4 in GAP, Magma, Sage, TeX

C_2\times C_8._{17}D_4
% in TeX

G:=Group("C2xC8.17D4");
// GroupNames label

G:=SmallGroup(128,879);
// by ID

G=gap.SmallGroup(128,879);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,1466,136,1411,172,4037,2028,124]);
// Polycyclic

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

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