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## G = C2×C8⋊8D4order 128 = 27

### Direct product of C2 and C8⋊8D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊8D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C2×C8⋊8D4
 Lower central C1 — C2 — C2×C4 — C2×C8⋊8D4
 Upper central C1 — C23 — C23×C4 — C2×C8⋊8D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊8D4

Generators and relations for C2×C88D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b3, dcd=c-1 >

Subgroups: 564 in 282 conjugacy classes, 116 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], D4 [×14], Q8 [×6], C23, C23 [×6], C23 [×12], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×10], SD16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C2×Q8 [×2], C2×Q8 [×5], C24, C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C88D4 [×8], C2×C4⋊D4, C2×C22⋊Q8, C23×C8, C22×SD16, C2×C88D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], SD16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×SD16 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C88D4 [×4], C2×C4⋊D4, C22×SD16, C2×C4○D8, C2×C88D4

Smallest permutation representation of C2×C88D4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(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 9 23 45)(2 12 24 48)(3 15 17 43)(4 10 18 46)(5 13 19 41)(6 16 20 44)(7 11 21 47)(8 14 22 42)(25 56 58 40)(26 51 59 35)(27 54 60 38)(28 49 61 33)(29 52 62 36)(30 55 63 39)(31 50 64 34)(32 53 57 37)
(1 39)(2 34)(3 37)(4 40)(5 35)(6 38)(7 33)(8 36)(9 63)(10 58)(11 61)(12 64)(13 59)(14 62)(15 57)(16 60)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(25 46)(26 41)(27 44)(28 47)(29 42)(30 45)(31 48)(32 43)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A ··· 4H 4I ··· 4N 8A ··· 8P order 1 2 ··· 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 8 8 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 SD16 C4○D8 kernel C2×C8⋊8D4 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.Q8 C8⋊8D4 C2×C4⋊D4 C2×C22⋊Q8 C23×C8 C22×SD16 C2×C8 C22×C4 C24 C2×C4 C23 C22 # reps 1 1 1 1 8 1 1 1 1 4 3 1 4 8 8

Matrix representation of C2×C88D4 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 5 0 0 0 12 12 0 0 0 0 0 5 12 0 0 0 5 5
,
 16 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 16 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,12,5],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1] >;

C2×C88D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_8D_4
% in TeX

G:=Group("C2xC8:8D4");
// GroupNames label

G:=SmallGroup(128,1779);
// by ID

G=gap.SmallGroup(128,1779);
# by ID

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

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

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