Copied to
clipboard

G = C9×SD16order 144 = 24·32

Direct product of C9 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×SD16, C726C2, C82C18, D4.C18, C24.6C6, Q82C18, C18.15D4, C36.18C22, (Q8×C9)⋊4C2, C2.4(D4×C9), C3.(C3×SD16), C4.2(C2×C18), (C3×D4).3C6, (D4×C9).2C2, C6.15(C3×D4), (C3×SD16).C3, (C3×Q8).6C6, C12.18(C2×C6), SmallGroup(144,26)

Series: Derived Chief Lower central Upper central

C1C4 — C9×SD16
C1C2C6C12C36Q8×C9 — C9×SD16
C1C2C4 — C9×SD16
C1C18C36 — C9×SD16

Generators and relations for C9×SD16
 G = < a,b,c | a9=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C22
2C4
4C6
2C12
2C2×C6
4C18
2C36
2C2×C18

Smallest permutation representation of C9×SD16
On 72 points
Generators in S72
(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)
(1 66 59 25 51 28 41 11)(2 67 60 26 52 29 42 12)(3 68 61 27 53 30 43 13)(4 69 62 19 54 31 44 14)(5 70 63 20 46 32 45 15)(6 71 55 21 47 33 37 16)(7 72 56 22 48 34 38 17)(8 64 57 23 49 35 39 18)(9 65 58 24 50 36 40 10)
(10 36)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 69)(20 70)(21 71)(22 72)(23 64)(24 65)(25 66)(26 67)(27 68)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)

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

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

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

C9×SD16 is a maximal subgroup of   D72⋊C2  SD16⋊D9  SD163D9

63 conjugacy classes

class 1 2A2B3A3B4A4B6A6B6C6D8A8B9A···9F12A12B12C12D18A···18F18G···18L24A24B24C24D36A···36F36G···36L72A···72L
order12233446666889···91212121218···1818···182424242436···3636···3672···72
size11411241144221···122441···14···422222···24···42···2

63 irreducible representations

dim111111111111222222
type+++++
imageC1C2C2C2C3C6C6C6C9C18C18C18D4SD16C3×D4C3×SD16D4×C9C9×SD16
kernelC9×SD16C72D4×C9Q8×C9C3×SD16C24C3×D4C3×Q8SD16C8D4Q8C18C9C6C3C2C1
# reps1111222266661224612

Matrix representation of C9×SD16 in GL2(𝔽19) generated by

90
09
,
05
46
,
111
018
G:=sub<GL(2,GF(19))| [9,0,0,9],[0,4,5,6],[1,0,11,18] >;

C9×SD16 in GAP, Magma, Sage, TeX

C_9\times {\rm SD}_{16}
% in TeX

G:=Group("C9xSD16");
// GroupNames label

G:=SmallGroup(144,26);
// by ID

G=gap.SmallGroup(144,26);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-2,432,169,122,2019,1017,165]);
// Polycyclic

G:=Group<a,b,c|a^9=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

Export

Subgroup lattice of C9×SD16 in TeX

׿
×
𝔽