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## G = C8⋊14SD16order 128 = 27

### 2nd semidirect product of C8 and SD16 acting via SD16/Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8⋊14SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8×Q8 — C8⋊14SD16
 Lower central C1 — C22 — C42 — C8⋊14SD16
 Upper central C1 — C22 — C42 — C8⋊14SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊14SD16

Generators and relations for C814SD16
G = < a,b,c | a8=b8=c2=1, bab-1=cac=a3, cbc=b3 >

Subgroups: 216 in 86 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×Q8, C41D4, C4⋊Q8, C2×SD16, C4.D8, C4.10D8, C82C8, C8×Q8, C4⋊SD16, Q8⋊Q8, C85D4, C814SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C4○D8, C8⋊C22, C4⋊SD16, C88D4, D4.3D4, C814SD16

Character table of C814SD16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N size 1 1 1 1 16 2 2 2 2 4 4 4 4 4 16 2 2 2 2 4 4 4 4 4 4 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 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 -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 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 -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 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 -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 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 -1 1 -1 linear of order 2 ρ9 2 2 2 2 0 -2 2 -2 2 0 0 -2 0 0 0 -2 -2 -2 -2 0 0 0 2 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -2 2 -2 2 0 0 -2 0 0 0 2 2 2 2 0 0 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 2 -2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 2 -2 2 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 -2i -2i 2i 0 0 2i 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 2 2 0 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 2i 2i -2i 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ16 2 2 -2 -2 0 -2 0 2 0 2i 0 0 -2i 0 0 -√-2 √-2 -√-2 √-2 -√2 √2 -√2 -√-2 √-2 √2 0 0 0 0 complex lifted from C4○D8 ρ17 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ18 2 2 -2 -2 0 2 0 -2 0 0 2 0 0 -2 0 -√-2 √-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ19 2 2 -2 -2 0 2 0 -2 0 0 -2 0 0 2 0 √-2 -√-2 √-2 -√-2 √-2 -√-2 -√-2 -√-2 √-2 √-2 0 0 0 0 complex lifted from SD16 ρ20 2 2 -2 -2 0 -2 0 2 0 -2i 0 0 2i 0 0 √-2 -√-2 √-2 -√-2 -√2 √2 -√2 √-2 -√-2 √2 0 0 0 0 complex lifted from C4○D8 ρ21 2 2 -2 -2 0 -2 0 2 0 -2i 0 0 2i 0 0 -√-2 √-2 -√-2 √-2 √2 -√2 √2 -√-2 √-2 -√2 0 0 0 0 complex lifted from C4○D8 ρ22 2 2 -2 -2 0 -2 0 2 0 2i 0 0 -2i 0 0 √-2 -√-2 √-2 -√-2 √2 -√2 √2 √-2 -√-2 -√2 0 0 0 0 complex lifted from C4○D8 ρ23 2 2 -2 -2 0 2 0 -2 0 0 2 0 0 -2 0 √-2 -√-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ24 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ25 2 2 -2 -2 0 2 0 -2 0 0 -2 0 0 2 0 -√-2 √-2 -√-2 √-2 -√-2 √-2 √-2 √-2 -√-2 -√-2 0 0 0 0 complex lifted from SD16 ρ26 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ27 4 -4 4 -4 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 2√-2 2√-2 -2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ29 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -2√-2 -2√-2 2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4

Smallest permutation representation of C814SD16
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 42 57 51 23 12 34 25)(2 45 58 54 24 15 35 28)(3 48 59 49 17 10 36 31)(4 43 60 52 18 13 37 26)(5 46 61 55 19 16 38 29)(6 41 62 50 20 11 39 32)(7 44 63 53 21 14 40 27)(8 47 64 56 22 9 33 30)
(2 4)(3 7)(6 8)(9 32)(10 27)(11 30)(12 25)(13 28)(14 31)(15 26)(16 29)(17 21)(18 24)(20 22)(33 62)(34 57)(35 60)(36 63)(37 58)(38 61)(39 64)(40 59)(41 56)(42 51)(43 54)(44 49)(45 52)(46 55)(47 50)(48 53)

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

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

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

Matrix representation of C814SD16 in GL4(𝔽17) generated by

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

C814SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_{14}{\rm SD}_{16}
% in TeX

G:=Group("C8:14SD16");
// GroupNames label

G:=SmallGroup(128,398);
// by ID

G=gap.SmallGroup(128,398);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,352,1123,136,2804,718,172]);
// Polycyclic

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

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