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## G = C3×C33⋊5C4order 324 = 22·34

### Direct product of C3 and C33⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3×C33⋊5C4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C33×C6 — C3×C33⋊5C4
 Lower central C33 — C3×C33⋊5C4
 Upper central C1 — C6

Generators and relations for C3×C335C4
G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 792 in 324 conjugacy classes, 114 normal (10 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, Dic3, C12, C3×C6, C3×C6, C33, C33, C33, C3×Dic3, C3⋊Dic3, C32×C6, C32×C6, C32×C6, C34, C3×C3⋊Dic3, C335C4, C33×C6, C3×C335C4
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, C3×C3⋊S3, C33⋊C2, C3×C3⋊Dic3, C335C4, C3×C33⋊C2, C3×C335C4

Smallest permutation representation of C3×C335C4
On 108 points
Generators in S108
(1 90 34)(2 91 35)(3 92 36)(4 89 33)(5 25 78)(6 26 79)(7 27 80)(8 28 77)(9 30 71)(10 31 72)(11 32 69)(12 29 70)(13 49 74)(14 50 75)(15 51 76)(16 52 73)(17 21 94)(18 22 95)(19 23 96)(20 24 93)(37 62 68)(38 63 65)(39 64 66)(40 61 67)(41 45 99)(42 46 100)(43 47 97)(44 48 98)(53 88 106)(54 85 107)(55 86 108)(56 87 105)(57 104 82)(58 101 83)(59 102 84)(60 103 81)
(1 54 20)(2 17 55)(3 56 18)(4 19 53)(5 51 69)(6 70 52)(7 49 71)(8 72 50)(9 27 74)(10 75 28)(11 25 76)(12 73 26)(13 30 80)(14 77 31)(15 32 78)(16 79 29)(21 86 91)(22 92 87)(23 88 89)(24 90 85)(33 96 106)(34 107 93)(35 94 108)(36 105 95)(37 98 57)(38 58 99)(39 100 59)(40 60 97)(41 63 101)(42 102 64)(43 61 103)(44 104 62)(45 65 83)(46 84 66)(47 67 81)(48 82 68)
(1 49 60)(2 57 50)(3 51 58)(4 59 52)(5 38 18)(6 19 39)(7 40 20)(8 17 37)(9 43 85)(10 86 44)(11 41 87)(12 88 42)(13 81 34)(14 35 82)(15 83 36)(16 33 84)(21 62 28)(22 25 63)(23 64 26)(24 27 61)(29 106 46)(30 47 107)(31 108 48)(32 45 105)(53 100 70)(54 71 97)(55 98 72)(56 69 99)(65 95 78)(66 79 96)(67 93 80)(68 77 94)(73 89 102)(74 103 90)(75 91 104)(76 101 92)
(1 80 43)(2 44 77)(3 78 41)(4 42 79)(5 45 92)(6 89 46)(7 47 90)(8 91 48)(9 60 93)(10 94 57)(11 58 95)(12 96 59)(13 61 54)(14 55 62)(15 63 56)(16 53 64)(17 104 31)(18 32 101)(19 102 29)(20 30 103)(21 82 72)(22 69 83)(23 84 70)(24 71 81)(25 99 36)(26 33 100)(27 97 34)(28 35 98)(37 75 108)(38 105 76)(39 73 106)(40 107 74)(49 67 85)(50 86 68)(51 65 87)(52 88 66)
(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)(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)

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 3C ··· 3AO 4A 4B 6A 6B 6C ··· 6AO 12A 12B 12C 12D order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 12 12 12 12 size 1 1 1 1 2 ··· 2 27 27 1 1 2 ··· 2 27 27 27 27

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 kernel C3×C33⋊5C4 C33×C6 C33⋊5C4 C34 C32×C6 C33 C32×C6 C33 C3×C6 C32 # reps 1 1 2 2 2 4 13 13 26 26

Matrix representation of C3×C335C4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 9 5 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 3 0 0 0 0 0 0 9
,
 9 5 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 9 0 0 0 0 0 0 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 11 3 0 0 0 0 0 0 9 0 0 0 0 0 0 3
,
 10 3 0 0 0 0 1 3 0 0 0 0 0 0 7 8 0 0 0 0 7 6 0 0 0 0 0 0 0 1 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,0,0,0,0,0,5,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[9,0,0,0,0,0,5,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,11,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[10,1,0,0,0,0,3,3,0,0,0,0,0,0,7,7,0,0,0,0,8,6,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C3×C335C4 in GAP, Magma, Sage, TeX

C_3\times C_3^3\rtimes_5C_4
% in TeX

G:=Group("C3xC3^3:5C4");
// GroupNames label

G:=SmallGroup(324,157);
// by ID

G=gap.SmallGroup(324,157);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,579,2164,7781]);
// Polycyclic

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

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