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## G = C3×C32⋊7D8order 432 = 24·33

### Direct product of C3 and C32⋊7D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊7D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C12⋊S3 — C3×C32⋊7D8
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊7D8
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×C327D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 676 in 196 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×D4, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, D4⋊S3, C3×D8, C3×C3⋊S3, C32×C6, C32×C6, C3×C3⋊C8, C324C8, C3×D12, C12⋊S3, D4×C32, D4×C32, D4×C32, C32×C12, C6×C3⋊S3, C3×C62, C3×D4⋊S3, C327D8, C3×C324C8, C3×C12⋊S3, D4×C33, C3×C327D8
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊S3, C3⋊D4, C3×D4, S3×C6, C2×C3⋊S3, D4⋊S3, C3×D8, C3×C3⋊S3, C3×C3⋊D4, C327D4, C6×C3⋊S3, C3×D4⋊S3, C327D8, C3×C327D4, C3×C327D8

Smallest permutation representation of C3×C327D8
On 72 points
Generators in S72
(1 22 34)(2 23 35)(3 24 36)(4 17 37)(5 18 38)(6 19 39)(7 20 40)(8 21 33)(9 46 51)(10 47 52)(11 48 53)(12 41 54)(13 42 55)(14 43 56)(15 44 49)(16 45 50)(25 61 66)(26 62 67)(27 63 68)(28 64 69)(29 57 70)(30 58 71)(31 59 72)(32 60 65)
(1 68 11)(2 12 69)(3 70 13)(4 14 71)(5 72 15)(6 16 65)(7 66 9)(8 10 67)(17 43 30)(18 31 44)(19 45 32)(20 25 46)(21 47 26)(22 27 48)(23 41 28)(24 29 42)(33 52 62)(34 63 53)(35 54 64)(36 57 55)(37 56 58)(38 59 49)(39 50 60)(40 61 51)
(1 34 22)(2 23 35)(3 36 24)(4 17 37)(5 38 18)(6 19 39)(7 40 20)(8 21 33)(9 51 46)(10 47 52)(11 53 48)(12 41 54)(13 55 42)(14 43 56)(15 49 44)(16 45 50)(25 66 61)(26 62 67)(27 68 63)(28 64 69)(29 70 57)(30 58 71)(31 72 59)(32 60 65)
(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 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 18)(19 24)(20 23)(21 22)(25 28)(26 27)(29 32)(30 31)(33 34)(35 40)(36 39)(37 38)(41 46)(42 45)(43 44)(47 48)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 64)(62 63)(65 70)(66 69)(67 68)(71 72)

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4 6A 6B 6C ··· 6N 6O ··· 6AN 6AO 6AP 8A 8B 12A 12B 12C ··· 12N 24A 24B 24C 24D order 1 2 2 2 3 3 3 ··· 3 4 6 6 6 ··· 6 6 ··· 6 6 6 8 8 12 12 12 ··· 12 24 24 24 24 size 1 1 4 36 1 1 2 ··· 2 2 1 1 2 ··· 2 4 ··· 4 36 36 18 18 2 2 4 ··· 4 18 18 18 18

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×D8 C3×C3⋊D4 D4⋊S3 C3×D4⋊S3 kernel C3×C32⋊7D8 C3×C32⋊4C8 C3×C12⋊S3 D4×C33 C32⋊7D8 C32⋊4C8 C12⋊S3 D4×C32 D4×C32 C32×C6 C3×C12 C33 C3×D4 C3×C6 C3×C6 C12 C32 C6 C32 C3 # reps 1 1 1 1 2 2 2 2 4 1 4 2 8 8 2 8 4 16 4 8

Matrix representation of C3×C327D8 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 66 0 0 0 0 59 66 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 48 0 0 0 0 38 41
,
 7 66 0 0 0 0 59 66 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 48 0 0 0 0 35 0

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,59,0,0,0,0,66,66,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,38,0,0,0,0,48,41],[7,59,0,0,0,0,66,66,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,35,0,0,0,0,48,0] >;

C3×C327D8 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_7D_8
% in TeX

G:=Group("C3xC3^2:7D8");
// GroupNames label

G:=SmallGroup(432,491);
// by ID

G=gap.SmallGroup(432,491);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,1011,514,80,4037,14118]);
// Polycyclic

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

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