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## G = C2×C8.2D4order 128 = 27

### Direct product of C2 and C8.2D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8.2D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C2×C8.2D4
 Lower central C1 — C2 — C2×C4 — C2×C8.2D4
 Upper central C1 — C23 — C2×C42 — C2×C8.2D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8.2D4

Generators and relations for C2×C8.2D4
G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b3, dcd=c3 >

Subgroups: 548 in 282 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×6], Q8 [×18], C23, C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×12], SD16 [×16], Q16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×6], C2×Q8 [×15], C24, C8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×8], C2×SD16 [×8], C2×Q16 [×8], C2×Q16 [×8], C22×D4, C22×Q8, C22×Q8 [×2], C2×C8⋊C4, C8.2D4 [×8], C2×C4.4D4, C2×C4⋊Q8, C22×SD16 [×2], C22×Q16 [×2], C2×C8.2D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C8.C22 [×4], C22×D4 [×3], C8.2D4 [×4], C2×C41D4, C2×C8.C22 [×2], C2×C8.2D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 C8.C22 kernel C2×C8.2D4 C2×C8⋊C4 C8.2D4 C2×C4.4D4 C2×C4⋊Q8 C22×SD16 C22×Q16 C42 C2×C8 C22×C4 C22 # reps 1 1 8 1 1 2 2 2 8 2 4

Matrix representation of C2×C8.2D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 2 11 8 9 0 0 0 0 16 0 9 0 0 0 0 0 6 7 16 1 0 0 0 0 1 10 7 16
,
 1 1 0 0 0 0 0 0 15 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 9 11 6 0 0 0 0 7 0 6 0 0 0 0 0 8 1 7 10 0 0 0 0 10 16 1 7
,
 1 0 0 0 0 0 0 0 15 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 16 0 0 0 0 0 1 16 0

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

C2×C8.2D4 in GAP, Magma, Sage, TeX

C_2\times C_8._2D_4
% in TeX

G:=Group("C2xC8.2D4");
// GroupNames label

G:=SmallGroup(128,1881);
// by ID

G=gap.SmallGroup(128,1881);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,184,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^3,d*c*d=c^3>;
// generators/relations

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