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## G = C32×C42⋊C3order 432 = 24·33

### Direct product of C32 and C42⋊C3

Aliases: C32×C42⋊C3, C42⋊C33, C1223C3, C62.6A4, (C4×C12)⋊C32, C22.(C32×A4), (C2×C6).11(C3×A4), SmallGroup(432,463)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C32×C42⋊C3
 Chief series C1 — C22 — C42 — C42⋊C3 — C3×C42⋊C3 — C32×C42⋊C3
 Lower central C42 — C32×C42⋊C3
 Upper central C1 — C32

Generators and relations for C32×C42⋊C3
G = < a,b,c,d,e | a3=b3=c4=d4=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, ede-1=c-1d2 >

Subgroups: 552 in 108 conjugacy classes, 40 normal (7 characteristic)
C1, C2, C3, C3, C4, C22, C6, C2×C4, C32, C32, C12, A4, C2×C6, C42, C3×C6, C2×C12, C33, C3×C12, C3×A4, C62, C42⋊C3, C4×C12, C6×C12, C32×A4, C3×C42⋊C3, C122, C32×C42⋊C3
Quotients: C1, C3, C32, A4, C33, C3×A4, C42⋊C3, C32×A4, C3×C42⋊C3, C32×C42⋊C3

Smallest permutation representation of C32×C42⋊C3
On 108 points
Generators in S108
(1 33 17)(2 34 18)(3 35 19)(4 36 20)(5 25 21)(6 26 22)(7 27 23)(8 28 24)(9 29 13)(10 30 14)(11 31 15)(12 32 16)(37 69 53)(38 70 54)(39 71 55)(40 72 56)(41 61 57)(42 62 58)(43 63 59)(44 64 60)(45 65 49)(46 66 50)(47 67 51)(48 68 52)(73 105 89)(74 106 90)(75 107 91)(76 108 92)(77 97 93)(78 98 94)(79 99 95)(80 100 96)(81 101 85)(82 102 86)(83 103 87)(84 104 88)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)(97 105 101)(98 106 102)(99 107 103)(100 108 104)
(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)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)(97 98 99 100)(101 102 103 104)(105 106 107 108)
(1 3 2 4)(5 7 6 8)(9 11 10 12)(13 15 14 16)(17 19 18 20)(21 23 22 24)(25 27 26 28)(29 31 30 32)(33 35 34 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)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)(97 99)(98 100)(101 103)(102 104)(105 107)(106 108)
(1 107 53)(2 105 55)(3 106 54)(4 108 56)(5 99 57)(6 97 59)(7 98 58)(8 100 60)(9 103 49)(10 101 51)(11 102 50)(12 104 52)(13 83 65)(14 81 67)(15 82 66)(16 84 68)(17 75 69)(18 73 71)(19 74 70)(20 76 72)(21 79 61)(22 77 63)(23 78 62)(24 80 64)(25 95 41)(26 93 43)(27 94 42)(28 96 44)(29 87 45)(30 85 47)(31 86 46)(32 88 48)(33 91 37)(34 89 39)(35 90 38)(36 92 40)

G:=sub<Sym(108)| (1,33,17)(2,34,18)(3,35,19)(4,36,20)(5,25,21)(6,26,22)(7,27,23)(8,28,24)(9,29,13)(10,30,14)(11,31,15)(12,32,16)(37,69,53)(38,70,54)(39,71,55)(40,72,56)(41,61,57)(42,62,58)(43,63,59)(44,64,60)(45,65,49)(46,66,50)(47,67,51)(48,68,52)(73,105,89)(74,106,90)(75,107,91)(76,108,92)(77,97,93)(78,98,94)(79,99,95)(80,100,96)(81,101,85)(82,102,86)(83,103,87)(84,104,88), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,105,101)(98,106,102)(99,107,103)(100,108,104), (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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108), (1,3,2,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32)(33,35,34,36)(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)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108), (1,107,53)(2,105,55)(3,106,54)(4,108,56)(5,99,57)(6,97,59)(7,98,58)(8,100,60)(9,103,49)(10,101,51)(11,102,50)(12,104,52)(13,83,65)(14,81,67)(15,82,66)(16,84,68)(17,75,69)(18,73,71)(19,74,70)(20,76,72)(21,79,61)(22,77,63)(23,78,62)(24,80,64)(25,95,41)(26,93,43)(27,94,42)(28,96,44)(29,87,45)(30,85,47)(31,86,46)(32,88,48)(33,91,37)(34,89,39)(35,90,38)(36,92,40)>;

G:=Group( (1,33,17)(2,34,18)(3,35,19)(4,36,20)(5,25,21)(6,26,22)(7,27,23)(8,28,24)(9,29,13)(10,30,14)(11,31,15)(12,32,16)(37,69,53)(38,70,54)(39,71,55)(40,72,56)(41,61,57)(42,62,58)(43,63,59)(44,64,60)(45,65,49)(46,66,50)(47,67,51)(48,68,52)(73,105,89)(74,106,90)(75,107,91)(76,108,92)(77,97,93)(78,98,94)(79,99,95)(80,100,96)(81,101,85)(82,102,86)(83,103,87)(84,104,88), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,105,101)(98,106,102)(99,107,103)(100,108,104), (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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108), (1,3,2,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32)(33,35,34,36)(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)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108), (1,107,53)(2,105,55)(3,106,54)(4,108,56)(5,99,57)(6,97,59)(7,98,58)(8,100,60)(9,103,49)(10,101,51)(11,102,50)(12,104,52)(13,83,65)(14,81,67)(15,82,66)(16,84,68)(17,75,69)(18,73,71)(19,74,70)(20,76,72)(21,79,61)(22,77,63)(23,78,62)(24,80,64)(25,95,41)(26,93,43)(27,94,42)(28,96,44)(29,87,45)(30,85,47)(31,86,46)(32,88,48)(33,91,37)(34,89,39)(35,90,38)(36,92,40) );

G=PermutationGroup([[(1,33,17),(2,34,18),(3,35,19),(4,36,20),(5,25,21),(6,26,22),(7,27,23),(8,28,24),(9,29,13),(10,30,14),(11,31,15),(12,32,16),(37,69,53),(38,70,54),(39,71,55),(40,72,56),(41,61,57),(42,62,58),(43,63,59),(44,64,60),(45,65,49),(46,66,50),(47,67,51),(48,68,52),(73,105,89),(74,106,90),(75,107,91),(76,108,92),(77,97,93),(78,98,94),(79,99,95),(80,100,96),(81,101,85),(82,102,86),(83,103,87),(84,104,88)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92),(97,105,101),(98,106,102),(99,107,103),(100,108,104)], [(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),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96),(97,98,99,100),(101,102,103,104),(105,106,107,108)], [(1,3,2,4),(5,7,6,8),(9,11,10,12),(13,15,14,16),(17,19,18,20),(21,23,22,24),(25,27,26,28),(29,31,30,32),(33,35,34,36),(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),(73,75),(74,76),(77,79),(78,80),(81,83),(82,84),(85,87),(86,88),(89,91),(90,92),(93,95),(94,96),(97,99),(98,100),(101,103),(102,104),(105,107),(106,108)], [(1,107,53),(2,105,55),(3,106,54),(4,108,56),(5,99,57),(6,97,59),(7,98,58),(8,100,60),(9,103,49),(10,101,51),(11,102,50),(12,104,52),(13,83,65),(14,81,67),(15,82,66),(16,84,68),(17,75,69),(18,73,71),(19,74,70),(20,76,72),(21,79,61),(22,77,63),(23,78,62),(24,80,64),(25,95,41),(26,93,43),(27,94,42),(28,96,44),(29,87,45),(30,85,47),(31,86,46),(32,88,48),(33,91,37),(34,89,39),(35,90,38),(36,92,40)]])

72 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Z 4A 4B 4C 4D 6A ··· 6H 12A ··· 12AF order 1 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 12 ··· 12 size 1 3 1 ··· 1 16 ··· 16 3 3 3 3 3 ··· 3 3 ··· 3

72 irreducible representations

 dim 1 1 1 3 3 3 3 type + + image C1 C3 C3 A4 C3×A4 C42⋊C3 C3×C42⋊C3 kernel C32×C42⋊C3 C3×C42⋊C3 C122 C62 C2×C6 C32 C3 # reps 1 24 2 1 8 4 32

Matrix representation of C32×C42⋊C3 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 7 12 0 0 0 0 9 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 12 0 0 0 0 0 0 12 0 0 0 0 4 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12
,
 8 7 0 0 0 0 8 5 3 0 0 0 2 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,7,9,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,0,4,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12],[8,8,2,0,0,0,7,5,12,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C32×C42⋊C3 in GAP, Magma, Sage, TeX

C_3^2\times C_4^2\rtimes C_3
% in TeX

G:=Group("C3^2xC4^2:C3");
// GroupNames label

G:=SmallGroup(432,463);
// by ID

G=gap.SmallGroup(432,463);
# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,1515,360,10399,102,9077,15882]);
// Polycyclic

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

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