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## G = (C2×C8).165D4order 128 = 27

### 133rd non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).165D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.36D4 — (C2×C8).165D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).165D4
 Upper central C1 — C22 — C22×C4 — (C2×C8).165D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).165D4

Generators and relations for (C2×C8).165D4
G = < a,b,c,d | a2=b8=d2=1, c4=b4, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab3, dcd=ab4c3 >

Subgroups: 216 in 99 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×5], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×4], Q8 [×2], C23, C23, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×C4○D4, C4.C42, C22.C42 [×2], (C22×C8)⋊C2, C23.36D4 [×2], C2×C4.Q8, (C2×C8).165D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C23.11D4, D4.3D4 [×2], (C2×C8).165D4

Character table of (C2×C8).165D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 2 2 2 2 8 8 8 8 8 4 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ7 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ9 2 2 2 2 -2 -2 0 2 2 -2 -2 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 2 2 -2 -2 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 -2 -2 2 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 2 -2 -2 2 2 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 -2 2 2 -2 0 -2 2 -2 2 0 -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 -2 2 -2 2 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 -2 2 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ16 2 -2 -2 2 2 -2 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 2i 0 0 -2i 0 complex lifted from C4○D4 ρ17 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 -2i 0 0 0 0 2i complex lifted from C4○D4 ρ18 2 -2 -2 2 2 -2 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 -2i 0 0 2i 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 -2 2 0 -2 2 2 -2 2i 0 0 0 -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 2 -2 0 -2 2 -2 2 0 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 2i 0 0 0 0 -2i complex lifted from C4○D4 ρ22 2 -2 -2 2 -2 2 0 -2 2 2 -2 -2i 0 0 0 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 0 0 -2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 -2√-2 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 0 0 2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 2√-2 0 0 0 0 0 0 0 complex lifted from D4.3D4

Smallest permutation representation of (C2×C8).165D4
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(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 45 27 59 5 41 31 63)(2 15 28 35 6 11 32 39)(3 43 29 57 7 47 25 61)(4 13 30 33 8 9 26 37)(10 53 38 23 14 49 34 19)(12 51 40 21 16 55 36 17)(18 48 52 62 22 44 56 58)(20 46 54 60 24 42 50 64)
(2 18)(4 20)(6 22)(8 24)(9 33)(10 61)(11 35)(12 63)(13 37)(14 57)(15 39)(16 59)(25 29)(26 54)(27 31)(28 56)(30 50)(32 52)(34 43)(36 45)(38 47)(40 41)(42 60)(44 62)(46 64)(48 58)(49 53)(51 55)

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

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

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

Matrix representation of (C2×C8).165D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 2 0 0 0 0 16 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 16 0 0
,
 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,1,0,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

(C2×C8).165D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{165}D_4
% in TeX

G:=Group("(C2xC8).165D4");
// GroupNames label

G:=SmallGroup(128,811);
// by ID

G=gap.SmallGroup(128,811);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,422,387,58,718,172,2028,1027,124]);
// Polycyclic

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

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