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## G = SD32⋊3C4order 128 = 27

### 1st semidirect product of SD32 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — SD32⋊3C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4×D8 — SD32⋊3C4
 Lower central C1 — C2 — C4 — C8 — SD32⋊3C4
 Upper central C1 — C22 — C42 — C4×C8 — SD32⋊3C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — SD32⋊3C4

Generators and relations for SD323C4
G = < a,b,c | a16=b2=c4=1, bab=a7, cac-1=a9, bc=cb >

Subgroups: 196 in 80 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C16, C42, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C2.D8, C2×C16, SD32, C4×D4, C4×Q8, C2×D8, C2×Q16, C165C4, C2.D16, C2.Q32, C163C4, C4×D8, C4×Q16, C2×SD32, SD323C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, C16⋊C22, Q32⋊C2, SD323C4

Smallest permutation representation of SD323C4
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 39)(2 46)(3 37)(4 44)(5 35)(6 42)(7 33)(8 40)(9 47)(10 38)(11 45)(12 36)(13 43)(14 34)(15 41)(16 48)(17 63)(18 54)(19 61)(20 52)(21 59)(22 50)(23 57)(24 64)(25 55)(26 62)(27 53)(28 60)(29 51)(30 58)(31 49)(32 56)
(1 23 39 57)(2 32 40 50)(3 25 41 59)(4 18 42 52)(5 27 43 61)(6 20 44 54)(7 29 45 63)(8 22 46 56)(9 31 47 49)(10 24 48 58)(11 17 33 51)(12 26 34 60)(13 19 35 53)(14 28 36 62)(15 21 37 55)(16 30 38 64)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 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 C4 D4 D4 C4○D4 D8 C4○D8 C16⋊C22 Q32⋊C2 kernel SD32⋊3C4 C16⋊5C4 C2.D16 C2.Q32 C16⋊3C4 C4×D8 C4×Q16 C2×SD32 SD32 C42 C2×C8 C8 C2×C4 C4 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 2 4 4 2 2

Matrix representation of SD323C4 in GL6(𝔽17)

 6 8 0 0 0 0 6 11 0 0 0 0 0 0 14 4 12 1 0 0 13 14 16 12 0 0 5 16 3 13 0 0 1 5 4 3
,
 16 0 0 0 0 0 10 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(17))| [6,6,0,0,0,0,8,11,0,0,0,0,0,0,14,13,5,1,0,0,4,14,16,5,0,0,12,16,3,4,0,0,1,12,13,3],[16,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

SD323C4 in GAP, Magma, Sage, TeX

{\rm SD}_{32}\rtimes_3C_4
% in TeX

G:=Group("SD32:3C4");
// GroupNames label

G:=SmallGroup(128,907);
// by ID

G=gap.SmallGroup(128,907);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^7,c*a*c^-1=a^9,b*c=c*b>;
// generators/relations

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