Copied to
clipboard

## G = SD16⋊11D4order 128 = 27

### 2nd semidirect product of SD16 and D4 acting through Inn(SD16)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — SD16⋊11D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C2×C4○D4 — C2×C4○D8 — SD16⋊11D4
 Lower central C1 — C2 — C2×C4 — SD16⋊11D4
 Upper central C1 — C22 — C4×D4 — SD16⋊11D4
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊11D4

Generators and relations for SD1611D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 512 in 247 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C41D4, C41D4, C4⋊Q8, C22×C8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C8×D4, C4×SD16, D4⋊D4, D4.7D4, C4⋊D8, C42Q16, C88D4, C85D4, D46D4, Q86D4, C2×C4○D8, SD1611D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, 2+ 1+4, D42, C2×C4○D8, D4○SD16, SD1611D4

Smallest permutation representation of SD1611D4
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 25)(2 28)(3 31)(4 26)(5 29)(6 32)(7 27)(8 30)(9 39)(10 34)(11 37)(12 40)(13 35)(14 38)(15 33)(16 36)(17 49)(18 52)(19 55)(20 50)(21 53)(22 56)(23 51)(24 54)(41 63)(42 58)(43 61)(44 64)(45 59)(46 62)(47 57)(48 60)
(1 12 23 62)(2 13 24 63)(3 14 17 64)(4 15 18 57)(5 16 19 58)(6 9 20 59)(7 10 21 60)(8 11 22 61)(25 40 51 46)(26 33 52 47)(27 34 53 48)(28 35 54 41)(29 36 55 42)(30 37 56 43)(31 38 49 44)(32 39 50 45)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 size 1 1 1 1 4 4 4 4 8 8 2 ··· 2 4 4 4 8 8 8 8 2 2 2 2 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D8 2+ 1+4 D4○SD16 kernel SD16⋊11D4 C8×D4 C4×SD16 D4⋊D4 D4.7D4 C4⋊D8 C4⋊2Q16 C8⋊8D4 C8⋊5D4 D4⋊6D4 Q8⋊6D4 C2×C4○D8 C22⋊C4 C4⋊C4 SD16 C2×D4 C4 C4 C2 # reps 1 1 1 2 2 1 1 2 1 1 1 2 2 1 4 1 8 1 2

Matrix representation of SD1611D4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 5 12 0 0 5 5
,
 16 0 0 0 0 16 0 0 0 0 3 3 0 0 3 14
,
 1 2 0 0 16 16 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 16 16 0 0 0 0 0 13 0 0 4 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,0,16,0,0,0,0,3,3,0,0,3,14],[1,16,0,0,2,16,0,0,0,0,16,0,0,0,0,16],[1,16,0,0,0,16,0,0,0,0,0,4,0,0,13,0] >;

SD1611D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_{11}D_4
% in TeX

G:=Group("SD16:11D4");
// GroupNames label

G:=SmallGroup(128,2016);
// by ID

G=gap.SmallGroup(128,2016);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,352,2019,346,2804,1411,375,172]);
// Polycyclic

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

׿
×
𝔽