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## G = C9×D4.S3order 432 = 24·33

### Direct product of C9 and D4.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C9×D4.S3
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×Dic6 — C9×D4.S3
 Lower central C3 — C6 — C12 — C9×D4.S3
 Upper central C1 — C18 — C36 — D4×C9

Generators and relations for C9×D4.S3
G = < a,b,c,d,e | a9=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 160 in 76 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, SD16, C18, C18, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×D4, C3×D4, C3×Q8, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, C62, D4.S3, C3×SD16, C3×C18, C3×C18, C72, D4×C9, D4×C9, Q8×C9, C3×C3⋊C8, C3×Dic6, D4×C32, C9×Dic3, C3×C36, C6×C18, C9×SD16, C3×D4.S3, C9×C3⋊C8, C9×Dic6, D4×C3×C9, C9×D4.S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, SD16, C18, C3×S3, C3⋊D4, C3×D4, C2×C18, S3×C6, D4.S3, C3×SD16, S3×C9, D4×C9, C3×C3⋊D4, S3×C18, C9×SD16, C3×D4.S3, C9×C3⋊D4, C9×D4.S3

Smallest permutation representation of C9×D4.S3
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 47 41 28)(2 48 42 29)(3 49 43 30)(4 50 44 31)(5 51 45 32)(6 52 37 33)(7 53 38 34)(8 54 39 35)(9 46 40 36)(10 62 66 22)(11 63 67 23)(12 55 68 24)(13 56 69 25)(14 57 70 26)(15 58 71 27)(16 59 72 19)(17 60 64 20)(18 61 65 21)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 64)(18 65)(37 52)(38 53)(39 54)(40 46)(41 47)(42 48)(43 49)(44 50)(45 51)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)
(1 70 41 14)(2 71 42 15)(3 72 43 16)(4 64 44 17)(5 65 45 18)(6 66 37 10)(7 67 38 11)(8 68 39 12)(9 69 40 13)(19 49 59 30)(20 50 60 31)(21 51 61 32)(22 52 62 33)(23 53 63 34)(24 54 55 35)(25 46 56 36)(26 47 57 28)(27 48 58 29)

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

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

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

108 conjugacy classes

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

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 D4 D6 SD16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 S3×C9 D4×C9 C3×C3⋊D4 S3×C18 C9×SD16 C9×C3⋊D4 D4.S3 C3×D4.S3 C9×D4.S3 kernel C9×D4.S3 C9×C3⋊C8 C9×Dic6 D4×C3×C9 C3×D4.S3 C3×C3⋊C8 C3×Dic6 D4×C32 D4.S3 C3⋊C8 Dic6 C3×D4 D4×C9 C3×C18 C36 C3×C9 C3×D4 C18 C3×C6 C12 C32 D4 C6 C6 C4 C3 C2 C9 C3 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 1 1 1 2 2 2 2 2 4 6 6 4 6 12 12 1 2 6

Matrix representation of C9×D4.S3 in GL4(𝔽73) generated by

 16 0 0 0 0 16 0 0 0 0 64 0 0 0 0 64
,
 72 0 0 0 0 72 0 0 0 0 1 72 0 0 2 72
,
 72 0 0 0 0 1 0 0 0 0 72 0 0 0 71 1
,
 8 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 12 67 0 0 12 61
G:=sub<GL(4,GF(73))| [16,0,0,0,0,16,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,1,2,0,0,72,72],[72,0,0,0,0,1,0,0,0,0,72,71,0,0,0,1],[8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,12,12,0,0,67,61] >;

C9×D4.S3 in GAP, Magma, Sage, TeX

C_9\times D_4.S_3
% in TeX

G:=Group("C9xD4.S3");
// GroupNames label

G:=SmallGroup(432,151);
// by ID

G=gap.SmallGroup(432,151);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,504,197,142,2355,1186,192,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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