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## G = C8⋊8SD16order 128 = 27

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

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

Aliases: C88SD16, C8211C2, C42.652C23, C83Q816C2, (C2×C8).273D4, C4.1(C2×SD16), C85D4.11C2, C2.5(C85D4), C4⋊Q8.77C22, (C4×C8).426C22, C41D4.40C22, C22.53(C41D4), (C2×C4).709(C2×D4), SmallGroup(128,437)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8⋊8SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C82 — C8⋊8SD16
 Lower central C1 — C22 — C42 — C8⋊8SD16
 Upper central C1 — C22 — C42 — C8⋊8SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊8SD16

Generators and relations for C88SD16
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a3, cbc=b3 >

Subgroups: 272 in 107 conjugacy classes, 48 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C4×C8, C4.Q8, C41D4, C4⋊Q8, C2×SD16, C82, C85D4, C83Q8, C88SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C41D4, C2×SD16, C85D4, C88SD16

Smallest permutation representation of C88SD16
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 29 58 54 12 45 17 33)(2 30 59 55 13 46 18 34)(3 31 60 56 14 47 19 35)(4 32 61 49 15 48 20 36)(5 25 62 50 16 41 21 37)(6 26 63 51 9 42 22 38)(7 27 64 52 10 43 23 39)(8 28 57 53 11 44 24 40)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 58)(18 61)(19 64)(20 59)(21 62)(22 57)(23 60)(24 63)(25 50)(26 53)(27 56)(28 51)(29 54)(30 49)(31 52)(32 55)(33 45)(34 48)(35 43)(36 46)(37 41)(38 44)(39 47)(40 42)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 4A ··· 4F 4G 4H 4I 8A ··· 8X order 1 2 2 2 2 4 ··· 4 4 4 4 8 ··· 8 size 1 1 1 1 16 2 ··· 2 16 16 16 2 ··· 2

38 irreducible representations

 dim 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 D4 SD16 kernel C8⋊8SD16 C82 C8⋊5D4 C8⋊3Q8 C2×C8 C8 # reps 1 1 3 3 6 24

Matrix representation of C88SD16 in GL4(𝔽17) generated by

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

C88SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("C8:8SD16");
// GroupNames label

G:=SmallGroup(128,437);
// by ID

G=gap.SmallGroup(128,437);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,100,1123,136,2804,172]);
// Polycyclic

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

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