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## G = C2×C8⋊5D4order 128 = 27

### Direct product of C2 and C8⋊5D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊5D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C2×C8⋊5D4
 Lower central C1 — C2 — C2×C4 — C2×C8⋊5D4
 Upper central C1 — C23 — C2×C42 — C2×C8⋊5D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊5D4

Generators and relations for C2×C85D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >

Subgroups: 724 in 328 conjugacy classes, 132 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×12], C4 [×4], C22, C22 [×6], C22 [×20], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×8], D4 [×28], Q8 [×12], C23, C23 [×16], C42 [×4], C4⋊C4 [×8], C2×C8 [×12], SD16 [×32], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×4], C2×Q8 [×10], C24 [×2], C4×C8 [×4], C2×C42, C2×C4⋊C4 [×2], C41D4 [×4], C41D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×16], C2×SD16 [×16], C22×D4 [×2], C22×D4 [×2], C22×Q8 [×2], C2×C4×C8, C85D4 [×8], C2×C41D4, C2×C4⋊Q8, C22×SD16 [×4], C2×C85D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], SD16 [×8], C2×D4 [×18], C24, C41D4 [×4], C2×SD16 [×12], C22×D4 [×3], C85D4 [×4], C2×C41D4, C22×SD16 [×2], C2×C85D4

Smallest permutation representation of C2×C85D4
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(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 34 63 13)(2 35 64 14)(3 36 57 15)(4 37 58 16)(5 38 59 9)(6 39 60 10)(7 40 61 11)(8 33 62 12)(17 26 56 42)(18 27 49 43)(19 28 50 44)(20 29 51 45)(21 30 52 46)(22 31 53 47)(23 32 54 48)(24 25 55 41)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 48)(18 43)(19 46)(20 41)(21 44)(22 47)(23 42)(24 45)(25 51)(26 54)(27 49)(28 52)(29 55)(30 50)(31 53)(32 56)(33 60)(34 63)(35 58)(36 61)(37 64)(38 59)(39 62)(40 57)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 SD16 kernel C2×C8⋊5D4 C2×C4×C8 C8⋊5D4 C2×C4⋊1D4 C2×C4⋊Q8 C22×SD16 C42 C2×C8 C22×C4 C2×C4 # reps 1 1 8 1 1 4 2 8 2 16

Matrix representation of C2×C85D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 5 0 0 0 0 12 12
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 15 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 15 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C2×C85D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_5D_4
% in TeX

G:=Group("C2xC8:5D4");
// GroupNames label

G:=SmallGroup(128,1875);
// by ID

G=gap.SmallGroup(128,1875);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,2804,172]);
// 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,b*c=c*b,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations

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