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## G = C4.SD16order 64 = 26

### 4th non-split extension by C4 of SD16 acting via SD16/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.SD16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4.SD16
 Lower central C1 — C2 — C2×C4 — C4.SD16
 Upper central C1 — C22 — C42 — C4.SD16
 Jennings C1 — C2 — C2 — C2×C4 — C4.SD16

Generators and relations for C4.SD16
G = < a,b,c | a4=b8=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b3 >

Character table of C4.SD16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 2 2 2 8 8 8 8 2 2 2 2 2 2 2 2 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 0 2 0 -2 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ12 2 2 -2 -2 0 0 0 -2 0 2 0 0 0 0 -√2 √2 √2 √2 -√2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ13 2 2 -2 -2 0 0 0 2 0 -2 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -2 -2 0 0 0 -2 0 2 0 0 0 0 √2 -√2 -√2 -√2 √2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 2 -2 0 0 -2 0 2 0 0 0 0 0 0 0 2i 0 2i 0 -2i -2i complex lifted from C4○D4 ρ16 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 -2i -2i 0 2i 0 2i 0 0 complex lifted from C4○D4 ρ17 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 2i 2i 0 -2i 0 -2i 0 0 complex lifted from C4○D4 ρ18 2 -2 2 -2 0 0 -2 0 2 0 0 0 0 0 0 0 -2i 0 -2i 0 2i 2i complex lifted from C4○D4 ρ19 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ20 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ21 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 √-2 complex lifted from SD16 ρ22 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 -√-2 complex lifted from SD16

Smallest permutation representation of C4.SD16
Regular action on 64 points
Generators in S64
(1 38 29 15)(2 39 30 16)(3 40 31 9)(4 33 32 10)(5 34 25 11)(6 35 26 12)(7 36 27 13)(8 37 28 14)(17 53 41 57)(18 54 42 58)(19 55 43 59)(20 56 44 60)(21 49 45 61)(22 50 46 62)(23 51 47 63)(24 52 48 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 54 29 58)(2 61 30 49)(3 52 31 64)(4 59 32 55)(5 50 25 62)(6 57 26 53)(7 56 27 60)(8 63 28 51)(9 48 40 24)(10 19 33 43)(11 46 34 22)(12 17 35 41)(13 44 36 20)(14 23 37 47)(15 42 38 18)(16 21 39 45)

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

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

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

Matrix representation of C4.SD16 in GL4(𝔽17) generated by

 0 16 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 3 14 0 0 3 3 0 0 0 0 5 5 0 0 12 5
,
 7 1 0 0 1 10 0 0 0 0 0 16 0 0 16 0
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[3,3,0,0,14,3,0,0,0,0,5,12,0,0,5,5],[7,1,0,0,1,10,0,0,0,0,0,16,0,0,16,0] >;

C4.SD16 in GAP, Magma, Sage, TeX

C_4.{\rm SD}_{16}
% in TeX

G:=Group("C4.SD16");
// GroupNames label

G:=SmallGroup(64,168);
// by ID

G=gap.SmallGroup(64,168);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,103,362,50,963,117,1444,88]);
// Polycyclic

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

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