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## G = C4×C9○He3order 324 = 22·34

### Direct product of C4 and C9○He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×C9○He3
 Chief series C1 — C3 — C6 — C18 — C3×C18 — C2×C9○He3 — C4×C9○He3
 Lower central C1 — C3 — C4×C9○He3
 Upper central C1 — C36 — C4×C9○He3

Generators and relations for C4×C9○He3
G = < a,b,c,d,e | a4=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >

Subgroups: 123 in 99 conjugacy classes, 87 normal (15 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C9, C32, C12, C12, C18, C18, C3×C6, C3×C9, He3, 3- 1+2, C36, C36, C3×C12, C3×C18, C2×He3, C2×3- 1+2, C9○He3, C3×C36, C4×He3, C4×3- 1+2, C2×C9○He3, C4×C9○He3
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, C33, C3×C12, C32×C6, C9○He3, C32×C12, C2×C9○He3, C4×C9○He3

Smallest permutation representation of C4×C9○He3
On 108 points
Generators in S108
(1 77 23 70)(2 78 24 71)(3 79 25 72)(4 80 26 64)(5 81 27 65)(6 73 19 66)(7 74 20 67)(8 75 21 68)(9 76 22 69)(10 55 102 48)(11 56 103 49)(12 57 104 50)(13 58 105 51)(14 59 106 52)(15 60 107 53)(16 61 108 54)(17 62 100 46)(18 63 101 47)(28 98 44 82)(29 99 45 83)(30 91 37 84)(31 92 38 85)(32 93 39 86)(33 94 40 87)(34 95 41 88)(35 96 42 89)(36 97 43 90)
(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)
(10 13 16)(11 14 17)(12 15 18)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 103 106)(101 104 107)(102 105 108)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)
(1 52 28)(2 53 29)(3 54 30)(4 46 31)(5 47 32)(6 48 33)(7 49 34)(8 50 35)(9 51 36)(10 94 73)(11 95 74)(12 96 75)(13 97 76)(14 98 77)(15 99 78)(16 91 79)(17 92 80)(18 93 81)(19 55 40)(20 56 41)(21 57 42)(22 58 43)(23 59 44)(24 60 45)(25 61 37)(26 62 38)(27 63 39)(64 100 85)(65 101 86)(66 102 87)(67 103 88)(68 104 89)(69 105 90)(70 106 82)(71 107 83)(72 108 84)

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

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

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

132 conjugacy classes

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

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C3 C3 C4 C6 C6 C6 C12 C12 C12 C9○He3 C2×C9○He3 C4×C9○He3 kernel C4×C9○He3 C2×C9○He3 C3×C36 C4×He3 C4×3- 1+2 C9○He3 C3×C18 C2×He3 C2×3- 1+2 C3×C9 He3 3- 1+2 C4 C2 C1 # reps 1 1 8 2 16 2 8 2 16 16 4 32 6 6 12

Matrix representation of C4×C9○He3 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 12 0 0 0 0 12 0 0 0 0 12
,
 10 0 0 0 0 1 8 7 0 0 10 0 0 0 0 26
,
 1 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 10 0 0 0 0 4 14 9 0 0 0 26 0 9 29 33
G:=sub<GL(4,GF(37))| [6,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[10,0,0,0,0,1,0,0,0,8,10,0,0,7,0,26],[1,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,4,0,9,0,14,0,29,0,9,26,33] >;

C4×C9○He3 in GAP, Magma, Sage, TeX

C_4\times C_9\circ {\rm He}_3
% in TeX

G:=Group("C4xC9oHe3");
// GroupNames label

G:=SmallGroup(324,108);
// by ID

G=gap.SmallGroup(324,108);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,-3,324,1034,165]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^9=c^3=e^3=1,d^1=b^6,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=b^3*c,d*e=e*d>;
// generators/relations

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