direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C4×C9○He3, C36.C32, He3.6C12, C12.3C33, 3- 1+2⋊4C12, (C3×C36)⋊5C3, (C3×C9)⋊12C12, C9.2(C3×C12), C18.3(C3×C6), (C3×C18).20C6, (C4×He3).2C3, C6.4(C32×C6), (C2×He3).15C6, C3.3(C32×C12), C32.5(C3×C12), (C3×C12).5C32, (C4×3- 1+2)⋊3C3, (C2×3- 1+2).7C6, C2.(C2×C9○He3), (C3×C6).12(C3×C6), (C2×C9○He3).3C2, SmallGroup(324,108)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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