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G = C3xC32:9SD16order 432 = 24·33

Direct product of C3 and C32:9SD16

direct product, metabelian, supersoluble, monomial

Aliases: C3xC32:9SD16, C33:23SD16, C12.33(S3xC6), C32:4Q8:9C6, C32:4C8:11C6, (C3xC12).128D6, (D4xC33).2C2, (C32xC6).74D4, (D4xC32).12C6, (D4xC32).15S3, C32:13(C3xSD16), C32:14(D4.S3), C6.36(C32:7D4), (C32xC12).28C22, D4.(C3xC3:S3), C4.2(C6xC3:S3), C3:3(C3xD4.S3), C12.53(C2xC3:S3), (C3xC6).70(C3xD4), C6.39(C3xC3:D4), (C3xC12).47(C2xC6), (C3xD4).8(C3:S3), (C3xD4).13(C3xS3), (C3xC32:4Q8):5C2, C2.5(C3xC32:7D4), (C3xC32:4C8):16C2, (C3xC6).109(C3:D4), SmallGroup(432,492)

Series: Derived Chief Lower central Upper central

C1C3xC12 — C3xC32:9SD16
C1C3C32C3xC6C3xC12C32xC12C3xC32:4Q8 — C3xC32:9SD16
C32C3xC6C3xC12 — C3xC32:9SD16
C1C6C12C3xD4

Generators and relations for C3xC32:9SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 500 in 184 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, C2xC6, SD16, C3xC6, C3xC6, C3xC6, C3:C8, C24, Dic6, C3xD4, C3xD4, C3xD4, C3xQ8, C33, C3xDic3, C3:Dic3, C3xC12, C3xC12, C3xC12, C62, D4.S3, C3xSD16, C32xC6, C32xC6, C3xC3:C8, C32:4C8, C3xDic6, C32:4Q8, D4xC32, D4xC32, D4xC32, C3xC3:Dic3, C32xC12, C3xC62, C3xD4.S3, C32:9SD16, C3xC32:4C8, C3xC32:4Q8, D4xC33, C3xC32:9SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, SD16, C3xS3, C3:S3, C3:D4, C3xD4, S3xC6, C2xC3:S3, D4.S3, C3xSD16, C3xC3:S3, C3xC3:D4, C32:7D4, C6xC3:S3, C3xD4.S3, C32:9SD16, C3xC32:7D4, C3xC32:9SD16

Smallest permutation representation of C3xC32:9SD16
On 72 points
Generators in S72
(1 27 61)(2 28 62)(3 29 63)(4 30 64)(5 31 57)(6 32 58)(7 25 59)(8 26 60)(9 35 43)(10 36 44)(11 37 45)(12 38 46)(13 39 47)(14 40 48)(15 33 41)(16 34 42)(17 68 51)(18 69 52)(19 70 53)(20 71 54)(21 72 55)(22 65 56)(23 66 49)(24 67 50)
(1 24 47)(2 48 17)(3 18 41)(4 42 19)(5 20 43)(6 44 21)(7 22 45)(8 46 23)(9 31 71)(10 72 32)(11 25 65)(12 66 26)(13 27 67)(14 68 28)(15 29 69)(16 70 30)(33 63 52)(34 53 64)(35 57 54)(36 55 58)(37 59 56)(38 49 60)(39 61 50)(40 51 62)
(1 61 27)(2 28 62)(3 63 29)(4 30 64)(5 57 31)(6 32 58)(7 59 25)(8 26 60)(9 43 35)(10 36 44)(11 45 37)(12 38 46)(13 47 39)(14 40 48)(15 41 33)(16 34 42)(17 68 51)(18 52 69)(19 70 53)(20 54 71)(21 72 55)(22 56 65)(23 66 49)(24 50 67)
(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)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)

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

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

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

81 conjugacy classes

class 1 2A2B3A3B3C···3N4A4B6A6B6C···6N6O···6AN8A8B12A12B12C···12N12O12P24A24B24C24D
order122333···344666···66···688121212···12121224242424
size114112···2236112···24···41818224···4363618181818

81 irreducible representations

dim11111111222222222244
type+++++++-
imageC1C2C2C2C3C6C6C6S3D4D6SD16C3xS3C3:D4C3xD4S3xC6C3xSD16C3xC3:D4D4.S3C3xD4.S3
kernelC3xC32:9SD16C3xC32:4C8C3xC32:4Q8D4xC33C32:9SD16C32:4C8C32:4Q8D4xC32D4xC32C32xC6C3xC12C33C3xD4C3xC6C3xC6C12C32C6C32C3
# reps111122224142882841648

Matrix representation of C3xC32:9SD16 in GL6(F73)

800000
080000
001000
000100
000010
000001
,
100000
010000
0064000
000800
000010
000001
,
6400000
080000
001000
000100
000010
000001
,
010000
7200000
000100
0072000
00006160
0000280
,
100000
0720000
001000
0007200
000010
00004472

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,61,28,0,0,0,0,60,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,44,0,0,0,0,0,72] >;

C3xC32:9SD16 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC3^2:9SD16");
// GroupNames label

G:=SmallGroup(432,492);
// by ID

G=gap.SmallGroup(432,492);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,1011,514,80,4037,14118]);
// Polycyclic

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

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