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G = C329SD16order 144 = 24·32

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

metabelian, supersoluble, monomial

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

C1C3×C12 — C329SD16
C1C3C32C3×C6C3×C12C324Q8 — C329SD16
C32C3×C6C3×C12 — C329SD16
C1C2C4D4

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 [×4], C4, C4, C22, C6 [×4], C6 [×4], C8, D4, Q8, C32, Dic3 [×4], C12 [×4], C2×C6 [×4], SD16, C3×C6, C3×C6, C3⋊C8 [×4], Dic6 [×4], C3×D4 [×4], C3⋊Dic3, C3×C12, C62, D4.S3 [×4], C324C8, C324Q8, D4×C32, C329SD16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], SD16, C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, D4.S3 [×4], C327D4, C329SD16

Character table of C329SD16

 class 12A2B3A3B3C3D4A4B6A6B6C6D6E6F6G6H6I6J6K6L8A8B12A12B12C12D
 size 114222223622224444444418184444
ρ1111111111111111111111111111    trivial
ρ211-111111-11111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ311-11111111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ411111111-1111111111111-1-11111    linear of order 2
ρ522-2-1-12-120-12-1-1-2111111-2002-1-1-1    orthogonal lifted from D6
ρ622-2-12-1-120-1-1-12111-21-21100-1-1-12    orthogonal lifted from D6
ρ7222-1-1-1220-1-12-1-12-1-12-1-1-100-12-1-1    orthogonal lifted from S3
ρ82202222-2022220000000000-2-2-2-2    orthogonal lifted from D4
ρ9222-1-12-120-12-1-12-1-1-1-1-1-12002-1-1-1    orthogonal lifted from S3
ρ10222-12-1-120-1-1-12-1-1-12-12-1-100-1-1-12    orthogonal lifted from S3
ρ1122-22-1-1-1202-1-1-111-2111-2100-1-12-1    orthogonal lifted from D6
ρ122222-1-1-1202-1-1-1-1-12-1-1-12-100-1-12-1    orthogonal lifted from S3
ρ1322-2-1-1-1220-1-12-11-211-211100-12-1-1    orthogonal lifted from D6
ρ14220-1-1-12-20-1-12-1--30-3--30-3--3-3001-211    complex lifted from C3⋊D4
ρ15220-1-12-1-20-12-1-10--3--3--3-3-3-3000-2111    complex lifted from C3⋊D4
ρ16220-12-1-1-20-1-1-12-3--3-30-30--3--300111-2    complex lifted from C3⋊D4
ρ17220-12-1-1-20-1-1-12--3-3--30--30-3-300111-2    complex lifted from C3⋊D4
ρ18220-1-1-12-20-1-12-1-30--3-30--3-3--3001-211    complex lifted from C3⋊D4
ρ19220-1-12-1-20-12-1-10-3-3-3--3--3--3000-2111    complex lifted from C3⋊D4
ρ202202-1-1-1-202-1-1-1--3--30-3-3--30-30011-21    complex lifted from C3⋊D4
ρ212202-1-1-1-202-1-1-1-3-30--3--3-30--30011-21    complex lifted from C3⋊D4
ρ222-20222200-2-2-2-200000000-2--20000    complex lifted from SD16
ρ232-20222200-2-2-2-200000000--2-20000    complex lifted from SD16
ρ244-404-2-2-200-422200000000000000    symplectic lifted from D4.S3, Schur index 2
ρ254-40-2-24-2002-42200000000000000    symplectic lifted from D4.S3, Schur index 2
ρ264-40-2-2-240022-4200000000000000    symplectic lifted from D4.S3, Schur index 2
ρ274-40-24-2-200222-400000000000000    symplectic lifted from D4.S3, Schur index 2

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

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

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

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

100000
010000
001000
000100
0000640
000008
,
100000
010000
000100
00727200
000080
0000064
,
0690000
18610000
0031300
00454200
000001
0000720
,
100000
3720000
001000
000100
000010
0000072

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

Export

Character table of C329SD16 in TeX

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