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## G = C2×D4⋊C8order 128 = 27

### Direct product of C2 and D4⋊C8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×D4⋊C8
G = < a,b,c,d | a2=b4=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 348 in 172 conjugacy classes, 76 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×4], C4 [×4], C22, C22 [×6], C22 [×16], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×14], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×10], C22×C4 [×3], C22×C4 [×9], C2×D4 [×6], C2×D4 [×3], C24, C4×C8 [×2], C4×C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C22×C8 [×2], C23×C4, C22×D4, D4⋊C8 [×4], C2×C4×C8, C2×C4⋊C8, C2×C4×D4, C2×D4⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], D4⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C22×C8, C2×M4(2), C2×D8, C2×SD16, D4⋊C8 [×4], C2×C22⋊C8, C2×D4⋊C4, C2×C4≀C2, C2×D4⋊C8

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 4Q 4R 4S 4T 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

56 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 C4 C4 C4 C8 D4 D4 M4(2) D8 SD16 C4≀C2 kernel C2×D4⋊C8 D4⋊C8 C2×C4×C8 C2×C4⋊C8 C2×C4×D4 C2×C4⋊C4 C4×D4 C22×D4 C2×D4 C42 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 2 4 2 16 2 2 4 4 4 8

Matrix representation of C2×D4⋊C8 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 16 0
,
 1 0 0 0 0 16 0 0 0 0 0 1 0 0 1 0
,
 9 0 0 0 0 4 0 0 0 0 3 14 0 0 14 14
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[9,0,0,0,0,4,0,0,0,0,3,14,0,0,14,14] >;

C2×D4⋊C8 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes C_8
% in TeX

G:=Group("C2xD4:C8");
// GroupNames label

G:=SmallGroup(128,206);
// by ID

G=gap.SmallGroup(128,206);
# by ID

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

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

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