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## 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

 Derived series C1 — C4 — C9×SD16
 Chief series C1 — C2 — C6 — C12 — C36 — Q8×C9 — C9×SD16
 Lower central C1 — C2 — C4 — C9×SD16
 Upper central C1 — C18 — C36 — C9×SD16

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

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

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

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

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

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

63 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 6C 6D 8A 8B 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 18G ··· 18L 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 2 3 3 4 4 6 6 6 6 8 8 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 4 1 1 2 4 1 1 4 4 2 2 1 ··· 1 2 2 4 4 1 ··· 1 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 D4 SD16 C3×D4 C3×SD16 D4×C9 C9×SD16 kernel C9×SD16 C72 D4×C9 Q8×C9 C3×SD16 C24 C3×D4 C3×Q8 SD16 C8 D4 Q8 C18 C9 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 1 2 2 4 6 12

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

 9 0 0 9
,
 0 5 4 6
,
 1 11 0 18
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

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