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## G = C23×C9⋊C6order 432 = 24·33

### Direct product of C23 and C9⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C23×C9⋊C6
 Chief series C1 — C3 — C9 — 3- 1+2 — C9⋊C6 — C2×C9⋊C6 — C22×C9⋊C6 — C23×C9⋊C6
 Lower central C9 — C23×C9⋊C6
 Upper central C1 — C23

Generators and relations for C23×C9⋊C6
G = < a,b,c,d,e | a2=b2=c2=d9=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d2 >

Subgroups: 1486 in 418 conjugacy classes, 182 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C9, C9, C32, D6, C2×C6, C2×C6, C24, D9, C18, C18, C3×S3, C3×C6, C22×S3, C22×C6, C22×C6, 3- 1+2, D18, C2×C18, C2×C18, S3×C6, C62, S3×C23, C23×C6, C9⋊C6, C2×3- 1+2, C22×D9, C22×C18, C22×C18, S3×C2×C6, C2×C62, C2×C9⋊C6, C22×3- 1+2, C23×D9, S3×C22×C6, C22×C9⋊C6, C23×3- 1+2, C23×C9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, S3×C6, S3×C23, C23×C6, C9⋊C6, S3×C2×C6, C2×C9⋊C6, S3×C22×C6, C22×C9⋊C6, C23×C9⋊C6

Smallest permutation representation of C23×C9⋊C6
On 72 points
Generators in S72
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 46)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 28)(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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)(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 29)(2 34 8 28 5 31)(3 30 6 36 9 33)(4 35)(7 32)(10 24 13 21 16 27)(11 20)(12 25 18 19 15 22)(14 26)(17 23)(37 69 40 66 43 72)(38 65)(39 70 45 64 42 67)(41 71)(44 68)(46 60 49 57 52 63)(47 56)(48 61 54 55 51 58)(50 62)(53 59)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 C9⋊C6 C2×C9⋊C6 kernel C23×C9⋊C6 C22×C9⋊C6 C23×3- 1+2 C23×D9 C22×D9 C22×C18 C2×C62 C62 C22×C6 C2×C6 C23 C22 # reps 1 14 1 2 28 2 1 7 2 14 1 7

Matrix representation of C23×C9⋊C6 in GL10(𝔽19)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18
,
 18 1 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 1 1 18 17 0 0 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 0 18 0 1 0 0 0 0 1 1 0 18 18 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 18 0 0
,
 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 18 18 0 0 0 0 1 1 18 18 0 0 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(10,GF(19))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18],[18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,17,18,18,18,18,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18,0,0],[0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0] >;

C23×C9⋊C6 in GAP, Magma, Sage, TeX

C_2^3\times C_9\rtimes C_6
% in TeX

G:=Group("C2^3xC9:C6");
// GroupNames label

G:=SmallGroup(432,559);
// by ID

G=gap.SmallGroup(432,559);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,10085,537,292,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^9=e^6=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^-1=d^2>;
// generators/relations

׿
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