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## G = C32⋊9SD16order 144 = 24·32

### 2nd semidirect product of C32 and SD16 acting via SD16/D4=C2

Aliases: C12.17D6, C329SD16, D4.(C3⋊S3), (C3×D4).5S3, (C3×C6).35D4, C33(D4.S3), C324C84C2, C324Q83C2, C6.23(C3⋊D4), (D4×C32).2C2, (C3×C12).13C22, C2.5(C327D4), C4.2(C2×C3⋊S3), SmallGroup(144,97)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C32⋊9SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4Q8 — C32⋊9SD16
 Lower central C32 — C3×C6 — C3×C12 — C32⋊9SD16
 Upper central C1 — C2 — C4 — D4

Generators and relations for C329SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd=c3 >

Subgroups: 170 in 60 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, Dic3, C12, C2×C6, SD16, C3×C6, C3×C6, C3⋊C8, Dic6, C3×D4, C3⋊Dic3, C3×C12, C62, D4.S3, C324C8, C324Q8, D4×C32, C329SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, C3⋊D4, C2×C3⋊S3, D4.S3, C327D4, C329SD16

Character table of C329SD16

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 12A 12B 12C 12D size 1 1 4 2 2 2 2 2 36 2 2 2 2 4 4 4 4 4 4 4 4 18 18 4 4 4 4 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 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 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 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 linear of order 2 ρ5 2 2 -2 -1 -1 2 -1 2 0 -1 2 -1 -1 -2 1 1 1 1 1 1 -2 0 0 2 -1 -1 -1 orthogonal lifted from D6 ρ6 2 2 -2 -1 2 -1 -1 2 0 -1 -1 -1 2 1 1 1 -2 1 -2 1 1 0 0 -1 -1 -1 2 orthogonal lifted from D6 ρ7 2 2 2 -1 -1 -1 2 2 0 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 0 0 -1 2 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 2 2 2 2 -2 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ9 2 2 2 -1 -1 2 -1 2 0 -1 2 -1 -1 2 -1 -1 -1 -1 -1 -1 2 0 0 2 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 -1 2 -1 -1 2 0 -1 -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 0 0 -1 -1 -1 2 orthogonal lifted from S3 ρ11 2 2 -2 2 -1 -1 -1 2 0 2 -1 -1 -1 1 1 -2 1 1 1 -2 1 0 0 -1 -1 2 -1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 -1 -1 2 0 2 -1 -1 -1 -1 -1 2 -1 -1 -1 2 -1 0 0 -1 -1 2 -1 orthogonal lifted from S3 ρ13 2 2 -2 -1 -1 -1 2 2 0 -1 -1 2 -1 1 -2 1 1 -2 1 1 1 0 0 -1 2 -1 -1 orthogonal lifted from D6 ρ14 2 2 0 -1 -1 -1 2 -2 0 -1 -1 2 -1 -√-3 0 √-3 -√-3 0 √-3 -√-3 √-3 0 0 1 -2 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 -1 -1 2 -1 -2 0 -1 2 -1 -1 0 -√-3 -√-3 -√-3 √-3 √-3 √-3 0 0 0 -2 1 1 1 complex lifted from C3⋊D4 ρ16 2 2 0 -1 2 -1 -1 -2 0 -1 -1 -1 2 √-3 -√-3 √-3 0 √-3 0 -√-3 -√-3 0 0 1 1 1 -2 complex lifted from C3⋊D4 ρ17 2 2 0 -1 2 -1 -1 -2 0 -1 -1 -1 2 -√-3 √-3 -√-3 0 -√-3 0 √-3 √-3 0 0 1 1 1 -2 complex lifted from C3⋊D4 ρ18 2 2 0 -1 -1 -1 2 -2 0 -1 -1 2 -1 √-3 0 -√-3 √-3 0 -√-3 √-3 -√-3 0 0 1 -2 1 1 complex lifted from C3⋊D4 ρ19 2 2 0 -1 -1 2 -1 -2 0 -1 2 -1 -1 0 √-3 √-3 √-3 -√-3 -√-3 -√-3 0 0 0 -2 1 1 1 complex lifted from C3⋊D4 ρ20 2 2 0 2 -1 -1 -1 -2 0 2 -1 -1 -1 -√-3 -√-3 0 √-3 √-3 -√-3 0 √-3 0 0 1 1 -2 1 complex lifted from C3⋊D4 ρ21 2 2 0 2 -1 -1 -1 -2 0 2 -1 -1 -1 √-3 √-3 0 -√-3 -√-3 √-3 0 -√-3 0 0 1 1 -2 1 complex lifted from C3⋊D4 ρ22 2 -2 0 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ23 2 -2 0 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ24 4 -4 0 4 -2 -2 -2 0 0 -4 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ25 4 -4 0 -2 -2 4 -2 0 0 2 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ26 4 -4 0 -2 -2 -2 4 0 0 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ27 4 -4 0 -2 4 -2 -2 0 0 2 2 2 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2

Smallest permutation representation of C329SD16
On 72 points
Generators in S72
(1 61 31)(2 32 62)(3 63 25)(4 26 64)(5 57 27)(6 28 58)(7 59 29)(8 30 60)(9 52 33)(10 34 53)(11 54 35)(12 36 55)(13 56 37)(14 38 49)(15 50 39)(16 40 51)(17 45 71)(18 72 46)(19 47 65)(20 66 48)(21 41 67)(22 68 42)(23 43 69)(24 70 44)
(1 15 45)(2 46 16)(3 9 47)(4 48 10)(5 11 41)(6 42 12)(7 13 43)(8 44 14)(17 31 39)(18 40 32)(19 25 33)(20 34 26)(21 27 35)(22 36 28)(23 29 37)(24 38 30)(49 60 70)(50 71 61)(51 62 72)(52 65 63)(53 64 66)(54 67 57)(55 58 68)(56 69 59)
(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)(65 66 67 68 69 70 71 72)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)

G:=sub<Sym(72)| (1,61,31)(2,32,62)(3,63,25)(4,26,64)(5,57,27)(6,28,58)(7,59,29)(8,30,60)(9,52,33)(10,34,53)(11,54,35)(12,36,55)(13,56,37)(14,38,49)(15,50,39)(16,40,51)(17,45,71)(18,72,46)(19,47,65)(20,66,48)(21,41,67)(22,68,42)(23,43,69)(24,70,44), (1,15,45)(2,46,16)(3,9,47)(4,48,10)(5,11,41)(6,42,12)(7,13,43)(8,44,14)(17,31,39)(18,40,32)(19,25,33)(20,34,26)(21,27,35)(22,36,28)(23,29,37)(24,38,30)(49,60,70)(50,71,61)(51,62,72)(52,65,63)(53,64,66)(54,67,57)(55,58,68)(56,69,59), (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)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)>;

G:=Group( (1,61,31)(2,32,62)(3,63,25)(4,26,64)(5,57,27)(6,28,58)(7,59,29)(8,30,60)(9,52,33)(10,34,53)(11,54,35)(12,36,55)(13,56,37)(14,38,49)(15,50,39)(16,40,51)(17,45,71)(18,72,46)(19,47,65)(20,66,48)(21,41,67)(22,68,42)(23,43,69)(24,70,44), (1,15,45)(2,46,16)(3,9,47)(4,48,10)(5,11,41)(6,42,12)(7,13,43)(8,44,14)(17,31,39)(18,40,32)(19,25,33)(20,34,26)(21,27,35)(22,36,28)(23,29,37)(24,38,30)(49,60,70)(50,71,61)(51,62,72)(52,65,63)(53,64,66)(54,67,57)(55,58,68)(56,69,59), (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)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70) );

G=PermutationGroup([[(1,61,31),(2,32,62),(3,63,25),(4,26,64),(5,57,27),(6,28,58),(7,59,29),(8,30,60),(9,52,33),(10,34,53),(11,54,35),(12,36,55),(13,56,37),(14,38,49),(15,50,39),(16,40,51),(17,45,71),(18,72,46),(19,47,65),(20,66,48),(21,41,67),(22,68,42),(23,43,69),(24,70,44)], [(1,15,45),(2,46,16),(3,9,47),(4,48,10),(5,11,41),(6,42,12),(7,13,43),(8,44,14),(17,31,39),(18,40,32),(19,25,33),(20,34,26),(21,27,35),(22,36,28),(23,29,37),(24,38,30),(49,60,70),(50,71,61),(51,62,72),(52,65,63),(53,64,66),(54,67,57),(55,58,68),(56,69,59)], [(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),(65,66,67,68,69,70,71,72)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(42,44),(43,47),(46,48),(49,55),(51,53),(52,56),(58,60),(59,63),(62,64),(65,69),(66,72),(68,70)]])

C329SD16 is a maximal subgroup of
S3×D4.S3  Dic6.19D6  D129D6  D12.22D6  C248D6  C24.26D6  SD16×C3⋊S3  C24.32D6  C62.131D4  C62.74D4  C62.75D4  He38SD16  C36.17D6  C3312SD16  C3314SD16  C3324SD16
C329SD16 is a maximal quotient of
C12.10Dic6  C62.114D4  C62.116D4  C36.17D6  He39SD16  C3312SD16  C3314SD16  C3324SD16

Matrix representation of C329SD16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 8 0 0 0 0 0 0 64
,
 0 69 0 0 0 0 18 61 0 0 0 0 0 0 31 3 0 0 0 0 45 42 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 3 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[0,18,0,0,0,0,69,61,0,0,0,0,0,0,31,45,0,0,0,0,3,42,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C329SD16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("C3^2:9SD16");
// GroupNames label

G:=SmallGroup(144,97);
// by ID

G=gap.SmallGroup(144,97);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,218,116,50,964,3461]);
// Polycyclic

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

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