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## G = C2×C8⋊D4order 128 = 27

### Direct product of C2 and C8⋊D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×C8⋊D4
 Lower central C1 — C2 — C2×C4 — C2×C8⋊D4
 Upper central C1 — C23 — C23×C4 — C2×C8⋊D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊D4

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

Subgroups: 564 in 272 conjugacy classes, 108 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], D4 [×14], Q8 [×6], C23, C23 [×2], C23 [×14], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×2], M4(2) [×8], 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], C2.D8 [×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], C2×M4(2) [×4], C2×M4(2) [×4], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C2×D4⋊C4, C2×Q8⋊C4, C2×C2.D8, C8⋊D4 [×8], C2×C4⋊D4, C2×C22⋊Q8, C22×M4(2), C22×SD16, C2×C8⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4 [×2], C2×C4○D4, C8⋊D4 [×4], C2×C4⋊D4, C2×C8⋊C22, C2×C8.C22, C2×C8⋊D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C2×C8⋊D4 C2×D4⋊C4 C2×Q8⋊C4 C2×C2.D8 C8⋊D4 C2×C4⋊D4 C2×C22⋊Q8 C22×M4(2) C22×SD16 C2×C8 C22×C4 C24 C2×C4 C22 C22 # reps 1 1 1 1 8 1 1 1 1 4 3 1 4 2 2

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

 1 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 14 3 0 11 0 0 0 0 3 14 11 6 0 0 0 0 3 14 0 0 0 0 0 0 3 11 14 6
,
 0 16 0 0 0 0 0 0 1 0 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 0 0 1 0 0 0 0 0 1 16 16 2 0 0 0 0 1 0 0 0 0 0 0 0 1 0 16 1
,
 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 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 16 0 0 0 0 0 1 0 16 1

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,14,3,3,3,0,0,0,0,3,14,14,11,0,0,0,0,0,11,0,14,0,0,0,0,11,6,0,6],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,16,0,0,0,0,0,2,0,1],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1] >;

C2×C8⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_4
% in TeX

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

G:=SmallGroup(128,1783);
// by ID

G=gap.SmallGroup(128,1783);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,2804,172]);
// 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=b^-1,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations

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