Copied to
clipboard

## G = C23×D9order 144 = 24·32

### Direct product of C23 and D9

Aliases: C23×D9, C9⋊C24, C18⋊C23, C3.(S3×C23), (C2×C6).35D6, (C2×C18)⋊4C22, (C22×C18)⋊3C2, C6.29(C22×S3), (C22×C6).10S3, SmallGroup(144,112)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C23×D9
 Chief series C1 — C3 — C9 — D9 — D18 — C22×D9 — C23×D9
 Lower central C9 — C23×D9
 Upper central C1 — C23

Generators and relations for C23×D9
G = < a,b,c,d,e | a2=b2=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 711 in 201 conjugacy classes, 99 normal (7 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C23, C9, D6, C2×C6, C24, D9, C18, C22×S3, C22×C6, D18, C2×C18, S3×C23, C22×D9, C22×C18, C23×D9
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C22×S3, D18, S3×C23, C22×D9, C23×D9

Smallest permutation representation of C23×D9
On 72 points
Generators in S72
(1 68)(2 69)(3 70)(4 71)(5 72)(6 64)(7 65)(8 66)(9 67)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 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 31)(2 30)(3 29)(4 28)(5 36)(6 35)(7 34)(8 33)(9 32)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(18 27)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 72)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(54 63)

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

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

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

C23×D9 is a maximal subgroup of   C223D36  C232D18
C23×D9 is a maximal quotient of   D46D18  Q8.15D18  D48D18  D4.10D18

48 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 6A ··· 6G 9A 9B 9C 18A ··· 18U order 1 2 ··· 2 2 ··· 2 3 6 ··· 6 9 9 9 18 ··· 18 size 1 1 ··· 1 9 ··· 9 2 2 ··· 2 2 2 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 S3 D6 D9 D18 kernel C23×D9 C22×D9 C22×C18 C22×C6 C2×C6 C23 C22 # reps 1 14 1 1 7 3 21

Matrix representation of C23×D9 in GL4(𝔽19) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 12 17 0 0 2 14
,
 18 0 0 0 0 1 0 0 0 0 2 7 0 0 5 17
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,12,2,0,0,17,14],[18,0,0,0,0,1,0,0,0,0,2,5,0,0,7,17] >;

C23×D9 in GAP, Magma, Sage, TeX

C_2^3\times D_9
% in TeX

G:=Group("C2^3xD9");
// GroupNames label

G:=SmallGroup(144,112);
// by ID

G=gap.SmallGroup(144,112);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,2404,208,3461]);
// Polycyclic

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

׿
×
𝔽