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## G = C4×He3.C3order 324 = 22·34

### Direct product of C4 and He3.C3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×He3.C3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — C2×He3.C3 — C4×He3.C3
 Lower central C1 — C3 — C32 — C4×He3.C3
 Upper central C1 — C12 — C3×C12 — C4×He3.C3

Generators and relations for C4×He3.C3
G = < a,b,c,d,e | a4=b3=c3=d3=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >

Smallest permutation representation of C4×He3.C3
On 108 points
Generators in S108
(1 64 10 98)(2 65 11 99)(3 66 12 91)(4 67 13 92)(5 68 14 93)(6 69 15 94)(7 70 16 95)(8 71 17 96)(9 72 18 97)(19 61 37 79)(20 62 38 80)(21 63 39 81)(22 55 40 73)(23 56 41 74)(24 57 42 75)(25 58 43 76)(26 59 44 77)(27 60 45 78)(28 102 48 82)(29 103 49 83)(30 104 50 84)(31 105 51 85)(32 106 52 86)(33 107 53 87)(34 108 54 88)(35 100 46 89)(36 101 47 90)
(1 62 29)(2 63 30)(3 55 31)(4 56 32)(5 57 33)(6 58 34)(7 59 35)(8 60 36)(9 61 28)(10 80 49)(11 81 50)(12 73 51)(13 74 52)(14 75 53)(15 76 54)(16 77 46)(17 78 47)(18 79 48)(19 82 97)(20 83 98)(21 84 99)(22 85 91)(23 86 92)(24 87 93)(25 88 94)(26 89 95)(27 90 96)(37 102 72)(38 103 64)(39 104 65)(40 105 66)(41 106 67)(42 107 68)(43 108 69)(44 100 70)(45 101 71)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(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 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)
(2 63 33)(3 31 61)(5 57 36)(6 34 55)(8 60 30)(9 28 58)(11 81 53)(12 51 79)(14 75 47)(15 54 73)(17 78 50)(18 48 76)(19 91 85)(20 23 26)(21 87 99)(22 94 88)(24 90 93)(25 97 82)(27 84 96)(29 35 32)(37 66 105)(38 41 44)(39 107 65)(40 69 108)(42 101 68)(43 72 102)(45 104 71)(46 52 49)(56 59 62)(74 77 80)(83 89 86)(100 106 103)
(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)| (1,64,10,98)(2,65,11,99)(3,66,12,91)(4,67,13,92)(5,68,14,93)(6,69,15,94)(7,70,16,95)(8,71,17,96)(9,72,18,97)(19,61,37,79)(20,62,38,80)(21,63,39,81)(22,55,40,73)(23,56,41,74)(24,57,42,75)(25,58,43,76)(26,59,44,77)(27,60,45,78)(28,102,48,82)(29,103,49,83)(30,104,50,84)(31,105,51,85)(32,106,52,86)(33,107,53,87)(34,108,54,88)(35,100,46,89)(36,101,47,90), (1,62,29)(2,63,30)(3,55,31)(4,56,32)(5,57,33)(6,58,34)(7,59,35)(8,60,36)(9,61,28)(10,80,49)(11,81,50)(12,73,51)(13,74,52)(14,75,53)(15,76,54)(16,77,46)(17,78,47)(18,79,48)(19,82,97)(20,83,98)(21,84,99)(22,85,91)(23,86,92)(24,87,93)(25,88,94)(26,89,95)(27,90,96)(37,102,72)(38,103,64)(39,104,65)(40,105,66)(41,106,67)(42,107,68)(43,108,69)(44,100,70)(45,101,71), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(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)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108), (2,63,33)(3,31,61)(5,57,36)(6,34,55)(8,60,30)(9,28,58)(11,81,53)(12,51,79)(14,75,47)(15,54,73)(17,78,50)(18,48,76)(19,91,85)(20,23,26)(21,87,99)(22,94,88)(24,90,93)(25,97,82)(27,84,96)(29,35,32)(37,66,105)(38,41,44)(39,107,65)(40,69,108)(42,101,68)(43,72,102)(45,104,71)(46,52,49)(56,59,62)(74,77,80)(83,89,86)(100,106,103), (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( (1,64,10,98)(2,65,11,99)(3,66,12,91)(4,67,13,92)(5,68,14,93)(6,69,15,94)(7,70,16,95)(8,71,17,96)(9,72,18,97)(19,61,37,79)(20,62,38,80)(21,63,39,81)(22,55,40,73)(23,56,41,74)(24,57,42,75)(25,58,43,76)(26,59,44,77)(27,60,45,78)(28,102,48,82)(29,103,49,83)(30,104,50,84)(31,105,51,85)(32,106,52,86)(33,107,53,87)(34,108,54,88)(35,100,46,89)(36,101,47,90), (1,62,29)(2,63,30)(3,55,31)(4,56,32)(5,57,33)(6,58,34)(7,59,35)(8,60,36)(9,61,28)(10,80,49)(11,81,50)(12,73,51)(13,74,52)(14,75,53)(15,76,54)(16,77,46)(17,78,47)(18,79,48)(19,82,97)(20,83,98)(21,84,99)(22,85,91)(23,86,92)(24,87,93)(25,88,94)(26,89,95)(27,90,96)(37,102,72)(38,103,64)(39,104,65)(40,105,66)(41,106,67)(42,107,68)(43,108,69)(44,100,70)(45,101,71), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(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)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108), (2,63,33)(3,31,61)(5,57,36)(6,34,55)(8,60,30)(9,28,58)(11,81,53)(12,51,79)(14,75,47)(15,54,73)(17,78,50)(18,48,76)(19,91,85)(20,23,26)(21,87,99)(22,94,88)(24,90,93)(25,97,82)(27,84,96)(29,35,32)(37,66,105)(38,41,44)(39,107,65)(40,69,108)(42,101,68)(43,72,102)(45,104,71)(46,52,49)(56,59,62)(74,77,80)(83,89,86)(100,106,103), (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([[(1,64,10,98),(2,65,11,99),(3,66,12,91),(4,67,13,92),(5,68,14,93),(6,69,15,94),(7,70,16,95),(8,71,17,96),(9,72,18,97),(19,61,37,79),(20,62,38,80),(21,63,39,81),(22,55,40,73),(23,56,41,74),(24,57,42,75),(25,58,43,76),(26,59,44,77),(27,60,45,78),(28,102,48,82),(29,103,49,83),(30,104,50,84),(31,105,51,85),(32,106,52,86),(33,107,53,87),(34,108,54,88),(35,100,46,89),(36,101,47,90)], [(1,62,29),(2,63,30),(3,55,31),(4,56,32),(5,57,33),(6,58,34),(7,59,35),(8,60,36),(9,61,28),(10,80,49),(11,81,50),(12,73,51),(13,74,52),(14,75,53),(15,76,54),(16,77,46),(17,78,47),(18,79,48),(19,82,97),(20,83,98),(21,84,99),(22,85,91),(23,86,92),(24,87,93),(25,88,94),(26,89,95),(27,90,96),(37,102,72),(38,103,64),(39,104,65),(40,105,66),(41,106,67),(42,107,68),(43,108,69),(44,100,70),(45,101,71)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(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),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108)], [(2,63,33),(3,31,61),(5,57,36),(6,34,55),(8,60,30),(9,28,58),(11,81,53),(12,51,79),(14,75,47),(15,54,73),(17,78,50),(18,48,76),(19,91,85),(20,23,26),(21,87,99),(22,94,88),(24,90,93),(25,97,82),(27,84,96),(29,35,32),(37,66,105),(38,41,44),(39,107,65),(40,69,108),(42,101,68),(43,72,102),(45,104,71),(46,52,49),(56,59,62),(74,77,80),(83,89,86),(100,106,103)], [(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)]])

68 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 9A ··· 9F 9G 9H 9I 9J 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 18A ··· 18F 18G 18H 18I 18J 36A ··· 36L 36M ··· 36T order 1 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 9 ··· 9 9 9 9 9 12 12 12 12 12 12 12 12 12 12 12 12 18 ··· 18 18 18 18 18 36 ··· 36 36 ··· 36 size 1 1 1 1 3 3 9 9 1 1 1 1 3 3 9 9 3 ··· 3 9 9 9 9 1 1 1 1 3 3 3 3 9 9 9 9 3 ··· 3 9 9 9 9 3 ··· 3 9 ··· 9

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C3 C4 C6 C6 C6 C12 C12 C12 He3 C2×He3 He3.C3 C4×He3 C2×He3.C3 C4×He3.C3 kernel C4×He3.C3 C2×He3.C3 C3×C36 C4×He3 C4×3- 1+2 He3.C3 C3×C18 C2×He3 C2×3- 1+2 C3×C9 He3 3- 1+2 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 4 2 2 2 4 4 4 8 2 2 6 4 6 12

Matrix representation of C4×He3.C3 in GL4(𝔽37) generated by

 6 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
,
 1 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 26 0 0 0 0 1 0 0 0 0 26 0 0 0 0 10
,
 10 0 0 0 0 28 25 28 0 28 28 25 0 25 28 28
G:=sub<GL(4,GF(37))| [6,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[26,0,0,0,0,1,0,0,0,0,26,0,0,0,0,10],[10,0,0,0,0,28,28,25,0,25,28,28,0,28,25,28] >;

C4×He3.C3 in GAP, Magma, Sage, TeX

C_4\times {\rm He}_3.C_3
% in TeX

G:=Group("C4xHe3.C3");
// GroupNames label

G:=SmallGroup(324,32);
// by ID

G=gap.SmallGroup(324,32);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,338,2170]);
// Polycyclic

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

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