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## G = C8.D4⋊C2order 128 = 27

### 3rd semidirect product of C8.D4 and C2 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C8.D4⋊C2
G = < a,b,c,d | a8=b4=d2=1, c2=a4, bab-1=a3, cac-1=a-1, ad=da, cbc-1=a4b-1, dbd=a4b, cd=dc >

Subgroups: 396 in 230 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×14], C23, C23 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×4], Q8⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22⋊Q8 [×8], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×Q16 [×4], C2×Q16 [×4], C22×Q8 [×2], C2×C4○D4, C23.38D4 [×2], C23.25D4, C8.18D4 [×4], C8.D4 [×4], C23.38C23 [×2], C2×C8○D4, C22×Q16, C8.D4⋊C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, Q8○D8 [×2], C8.D4⋊C2

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 Q8○D8 kernel C8.D4⋊C2 C23.38D4 C23.25D4 C8.18D4 C8.D4 C23.38C23 C2×C8○D4 C22×Q16 C2×C8 C2×D4 C2×Q8 C2×C4 C2 # reps 1 2 1 4 4 2 1 1 4 3 1 4 4

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

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 3 14 0 0 0 0 3 3 0 0
,
 3 4 0 0 0 0 6 14 0 0 0 0 0 0 10 16 0 0 0 0 16 7 0 0 0 0 0 0 7 1 0 0 0 0 1 10
,
 3 4 0 0 0 0 15 14 0 0 0 0 0 0 10 16 0 0 0 0 16 7 0 0 0 0 0 0 10 16 0 0 0 0 16 7
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,3,3,0,0,0,0,14,3,0,0],[3,6,0,0,0,0,4,14,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,1,10],[3,15,0,0,0,0,4,14,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C8.D4⋊C2 in GAP, Magma, Sage, TeX

C_8.D_4\rtimes C_2
% in TeX

G:=Group("C8.D4:C2");
// GroupNames label

G:=SmallGroup(128,1791);
// by ID

G=gap.SmallGroup(128,1791);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,521,2804,172]);
// Polycyclic

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

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