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## G = C3×He3⋊3C4order 324 = 22·34

### Direct product of C3 and He3⋊3C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C3×He3⋊3C4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C6×He3 — C3×He3⋊3C4
 Lower central He3 — C3×He3⋊3C4
 Upper central C1 — C3×C6

Generators and relations for C3×He33C4
G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 336 in 120 conjugacy classes, 34 normal (12 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, He3, He3, C33, C3×Dic3, C3×C12, C2×He3, C2×He3, C32×C6, C3×He3, He33C4, C32×Dic3, C6×He3, C3×He33C4
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, He3⋊C2, C3×C3⋊S3, He33C4, C3×C3⋊Dic3, C3×He3⋊C2, C3×He33C4

Smallest permutation representation of C3×He33C4
On 108 points
Generators in S108
(1 60 49)(2 57 50)(3 58 51)(4 59 52)(5 18 38)(6 19 39)(7 20 40)(8 17 37)(9 85 43)(10 86 44)(11 87 41)(12 88 42)(13 34 81)(14 35 82)(15 36 83)(16 33 84)(21 62 28)(22 63 25)(23 64 26)(24 61 27)(29 106 46)(30 107 47)(31 108 48)(32 105 45)(53 100 70)(54 97 71)(55 98 72)(56 99 69)(65 78 95)(66 79 96)(67 80 93)(68 77 94)(73 89 102)(74 90 103)(75 91 104)(76 92 101)
(1 49 60)(2 57 50)(3 51 58)(4 59 52)(5 105 101)(6 102 106)(7 107 103)(8 104 108)(9 43 85)(10 86 44)(11 41 87)(12 88 42)(13 71 27)(14 28 72)(15 69 25)(16 26 70)(17 75 48)(18 45 76)(19 73 46)(20 47 74)(21 55 35)(22 36 56)(23 53 33)(24 34 54)(29 39 89)(30 90 40)(31 37 91)(32 92 38)(61 81 97)(62 98 82)(63 83 99)(64 100 84)(65 95 78)(66 79 96)(67 93 80)(68 77 94)
(1 80 43)(2 77 44)(3 78 41)(4 79 42)(5 45 92)(6 46 89)(7 47 90)(8 48 91)(9 60 93)(10 57 94)(11 58 95)(12 59 96)(13 61 54)(14 62 55)(15 63 56)(16 64 53)(17 31 104)(18 32 101)(19 29 102)(20 30 103)(21 72 82)(22 69 83)(23 70 84)(24 71 81)(25 99 36)(26 100 33)(27 97 34)(28 98 35)(37 108 75)(38 105 76)(39 106 73)(40 107 74)(49 67 85)(50 68 86)(51 65 87)(52 66 88)
(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 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)| (1,60,49)(2,57,50)(3,58,51)(4,59,52)(5,18,38)(6,19,39)(7,20,40)(8,17,37)(9,85,43)(10,86,44)(11,87,41)(12,88,42)(13,34,81)(14,35,82)(15,36,83)(16,33,84)(21,62,28)(22,63,25)(23,64,26)(24,61,27)(29,106,46)(30,107,47)(31,108,48)(32,105,45)(53,100,70)(54,97,71)(55,98,72)(56,99,69)(65,78,95)(66,79,96)(67,80,93)(68,77,94)(73,89,102)(74,90,103)(75,91,104)(76,92,101), (1,49,60)(2,57,50)(3,51,58)(4,59,52)(5,105,101)(6,102,106)(7,107,103)(8,104,108)(9,43,85)(10,86,44)(11,41,87)(12,88,42)(13,71,27)(14,28,72)(15,69,25)(16,26,70)(17,75,48)(18,45,76)(19,73,46)(20,47,74)(21,55,35)(22,36,56)(23,53,33)(24,34,54)(29,39,89)(30,90,40)(31,37,91)(32,92,38)(61,81,97)(62,98,82)(63,83,99)(64,100,84)(65,95,78)(66,79,96)(67,93,80)(68,77,94), (1,80,43)(2,77,44)(3,78,41)(4,79,42)(5,45,92)(6,46,89)(7,47,90)(8,48,91)(9,60,93)(10,57,94)(11,58,95)(12,59,96)(13,61,54)(14,62,55)(15,63,56)(16,64,53)(17,31,104)(18,32,101)(19,29,102)(20,30,103)(21,72,82)(22,69,83)(23,70,84)(24,71,81)(25,99,36)(26,100,33)(27,97,34)(28,98,35)(37,108,75)(38,105,76)(39,106,73)(40,107,74)(49,67,85)(50,68,86)(51,65,87)(52,66,88), (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), (1,2,3,4)(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( (1,60,49)(2,57,50)(3,58,51)(4,59,52)(5,18,38)(6,19,39)(7,20,40)(8,17,37)(9,85,43)(10,86,44)(11,87,41)(12,88,42)(13,34,81)(14,35,82)(15,36,83)(16,33,84)(21,62,28)(22,63,25)(23,64,26)(24,61,27)(29,106,46)(30,107,47)(31,108,48)(32,105,45)(53,100,70)(54,97,71)(55,98,72)(56,99,69)(65,78,95)(66,79,96)(67,80,93)(68,77,94)(73,89,102)(74,90,103)(75,91,104)(76,92,101), (1,49,60)(2,57,50)(3,51,58)(4,59,52)(5,105,101)(6,102,106)(7,107,103)(8,104,108)(9,43,85)(10,86,44)(11,41,87)(12,88,42)(13,71,27)(14,28,72)(15,69,25)(16,26,70)(17,75,48)(18,45,76)(19,73,46)(20,47,74)(21,55,35)(22,36,56)(23,53,33)(24,34,54)(29,39,89)(30,90,40)(31,37,91)(32,92,38)(61,81,97)(62,98,82)(63,83,99)(64,100,84)(65,95,78)(66,79,96)(67,93,80)(68,77,94), (1,80,43)(2,77,44)(3,78,41)(4,79,42)(5,45,92)(6,46,89)(7,47,90)(8,48,91)(9,60,93)(10,57,94)(11,58,95)(12,59,96)(13,61,54)(14,62,55)(15,63,56)(16,64,53)(17,31,104)(18,32,101)(19,29,102)(20,30,103)(21,72,82)(22,69,83)(23,70,84)(24,71,81)(25,99,36)(26,100,33)(27,97,34)(28,98,35)(37,108,75)(38,105,76)(39,106,73)(40,107,74)(49,67,85)(50,68,86)(51,65,87)(52,66,88), (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), (1,2,3,4)(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([[(1,60,49),(2,57,50),(3,58,51),(4,59,52),(5,18,38),(6,19,39),(7,20,40),(8,17,37),(9,85,43),(10,86,44),(11,87,41),(12,88,42),(13,34,81),(14,35,82),(15,36,83),(16,33,84),(21,62,28),(22,63,25),(23,64,26),(24,61,27),(29,106,46),(30,107,47),(31,108,48),(32,105,45),(53,100,70),(54,97,71),(55,98,72),(56,99,69),(65,78,95),(66,79,96),(67,80,93),(68,77,94),(73,89,102),(74,90,103),(75,91,104),(76,92,101)], [(1,49,60),(2,57,50),(3,51,58),(4,59,52),(5,105,101),(6,102,106),(7,107,103),(8,104,108),(9,43,85),(10,86,44),(11,41,87),(12,88,42),(13,71,27),(14,28,72),(15,69,25),(16,26,70),(17,75,48),(18,45,76),(19,73,46),(20,47,74),(21,55,35),(22,36,56),(23,53,33),(24,34,54),(29,39,89),(30,90,40),(31,37,91),(32,92,38),(61,81,97),(62,98,82),(63,83,99),(64,100,84),(65,95,78),(66,79,96),(67,93,80),(68,77,94)], [(1,80,43),(2,77,44),(3,78,41),(4,79,42),(5,45,92),(6,46,89),(7,47,90),(8,48,91),(9,60,93),(10,57,94),(11,58,95),(12,59,96),(13,61,54),(14,62,55),(15,63,56),(16,64,53),(17,31,104),(18,32,101),(19,29,102),(20,30,103),(21,72,82),(22,69,83),(23,70,84),(24,71,81),(25,99,36),(26,100,33),(27,97,34),(28,98,35),(37,108,75),(38,105,76),(39,106,73),(40,107,74),(49,67,85),(50,68,86),(51,65,87),(52,66,88)], [(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)], [(1,2,3,4),(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)]])

60 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3T 4A 4B 6A ··· 6H 6I ··· 6T 12A ··· 12P order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 ··· 1 6 ··· 6 9 9 1 ··· 1 6 ··· 6 9 ··· 9

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 type + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 He3⋊C2 He3⋊3C4 kernel C3×He3⋊3C4 C6×He3 He3⋊3C4 C3×He3 C2×He3 He3 C32×C6 C33 C3×C6 C32 C6 C3 # reps 1 1 2 2 2 4 4 4 8 8 12 12

Matrix representation of C3×He33C4 in GL5(𝔽13)

 3 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 3 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 12 1 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 8 5 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 8 0 0 0 0 0 0 8

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

C3×He33C4 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3\rtimes_3C_4
% in TeX

G:=Group("C3xHe3:3C4");
// GroupNames label

G:=SmallGroup(324,99);
// by ID

G=gap.SmallGroup(324,99);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,579,2164,382]);
// 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,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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