direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C4×C3.He3, C12.5He3, 3- 1+2.C12, (C3×C9).3C12, (C3×C36).1C3, C6.6(C2×He3), C3.5(C4×He3), (C3×C18).14C6, C32.4(C3×C12), (C3×C12).4C32, (C4×3- 1+2).C3, (C2×3- 1+2).3C6, (C3×C6).5(C3×C6), C2.(C2×C3.He3), (C2×C3.He3).2C2, SmallGroup(324,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C3.He3
G = < a,b,c,d,e | a4=b3=d3=1, c3=b-1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=bcd-1, ede-1=b-1d >
Smallest permutation representation of C4×C3.He3
►On 108 points
Generators in S108
(1 81 27 67)(2 73 19 68)(3 74 20 69)(4 75 21 70)(5 76 22 71)(6 77 23 72)(7 78 24 64)(8 79 25 65)(9 80 26 66)(10 63 100 46)(11 55 101 47)(12 56 102 48)(13 57 103 49)(14 58 104 50)(15 59 105 51)(16 60 106 52)(17 61 107 53)(18 62 108 54)(28 97 43 82)(29 98 44 83)(30 99 45 84)(31 91 37 85)(32 92 38 86)(33 93 39 87)(34 94 40 88)(35 95 41 89)(36 96 42 90)
(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 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 49 28 7 46 34 4 52 31)(2 53 29 8 50 35 5 47 32)(3 48 30 9 54 36 6 51 33)(10 94 75 16 91 81 13 97 78)(11 92 73 17 98 79 14 95 76)(12 99 80 18 96 77 15 93 74)(19 61 44 25 58 41 22 55 38)(20 56 45 26 62 42 23 59 39)(21 60 37 27 57 43 24 63 40)(64 100 88 70 106 85 67 103 82)(65 104 89 71 101 86 68 107 83)(66 108 90 72 105 87 69 102 84)
G:=sub<Sym(108)| (1,81,27,67)(2,73,19,68)(3,74,20,69)(4,75,21,70)(5,76,22,71)(6,77,23,72)(7,78,24,64)(8,79,25,65)(9,80,26,66)(10,63,100,46)(11,55,101,47)(12,56,102,48)(13,57,103,49)(14,58,104,50)(15,59,105,51)(16,60,106,52)(17,61,107,53)(18,62,108,54)(28,97,43,82)(29,98,44,83)(30,99,45,84)(31,91,37,85)(32,92,38,86)(33,93,39,87)(34,94,40,88)(35,95,41,89)(36,96,42,90), (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), (1,2,3,4,5,6,7,8,9)(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), (1,49,28,7,46,34,4,52,31)(2,53,29,8,50,35,5,47,32)(3,48,30,9,54,36,6,51,33)(10,94,75,16,91,81,13,97,78)(11,92,73,17,98,79,14,95,76)(12,99,80,18,96,77,15,93,74)(19,61,44,25,58,41,22,55,38)(20,56,45,26,62,42,23,59,39)(21,60,37,27,57,43,24,63,40)(64,100,88,70,106,85,67,103,82)(65,104,89,71,101,86,68,107,83)(66,108,90,72,105,87,69,102,84)>;
G:=Group( (1,81,27,67)(2,73,19,68)(3,74,20,69)(4,75,21,70)(5,76,22,71)(6,77,23,72)(7,78,24,64)(8,79,25,65)(9,80,26,66)(10,63,100,46)(11,55,101,47)(12,56,102,48)(13,57,103,49)(14,58,104,50)(15,59,105,51)(16,60,106,52)(17,61,107,53)(18,62,108,54)(28,97,43,82)(29,98,44,83)(30,99,45,84)(31,91,37,85)(32,92,38,86)(33,93,39,87)(34,94,40,88)(35,95,41,89)(36,96,42,90), (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), (1,2,3,4,5,6,7,8,9)(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), (1,49,28,7,46,34,4,52,31)(2,53,29,8,50,35,5,47,32)(3,48,30,9,54,36,6,51,33)(10,94,75,16,91,81,13,97,78)(11,92,73,17,98,79,14,95,76)(12,99,80,18,96,77,15,93,74)(19,61,44,25,58,41,22,55,38)(20,56,45,26,62,42,23,59,39)(21,60,37,27,57,43,24,63,40)(64,100,88,70,106,85,67,103,82)(65,104,89,71,101,86,68,107,83)(66,108,90,72,105,87,69,102,84) );
G=PermutationGroup([[(1,81,27,67),(2,73,19,68),(3,74,20,69),(4,75,21,70),(5,76,22,71),(6,77,23,72),(7,78,24,64),(8,79,25,65),(9,80,26,66),(10,63,100,46),(11,55,101,47),(12,56,102,48),(13,57,103,49),(14,58,104,50),(15,59,105,51),(16,60,106,52),(17,61,107,53),(18,62,108,54),(28,97,43,82),(29,98,44,83),(30,99,45,84),(31,91,37,85),(32,92,38,86),(33,93,39,87),(34,94,40,88),(35,95,41,89),(36,96,42,90)], [(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)], [(1,2,3,4,5,6,7,8,9),(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)], [(1,49,28,7,46,34,4,52,31),(2,53,29,8,50,35,5,47,32),(3,48,30,9,54,36,6,51,33),(10,94,75,16,91,81,13,97,78),(11,92,73,17,98,79,14,95,76),(12,99,80,18,96,77,15,93,74),(19,61,44,25,58,41,22,55,38),(20,56,45,26,62,42,23,59,39),(21,60,37,27,57,43,24,63,40),(64,100,88,70,106,85,67,103,82),(65,104,89,71,101,86,68,107,83),(66,108,90,72,105,87,69,102,84)]])
68 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18F | 18G | ··· | 18L | 36A | ··· | 36L | 36M | ··· | 36X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 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 | 3 | ··· | 3 | 9 | ··· | 9 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 9 | ··· | 9 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | He3 | C2×He3 | C3.He3 | C4×He3 | C2×C3.He3 | C4×C3.He3 |
| kernel | C4×C3.He3 | C2×C3.He3 | C3×C36 | C4×3- 1+2 | C3.He3 | C3×C18 | C2×3- 1+2 | C3×C9 | 3- 1+2 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 2 | 2 | 6 | 4 | 12 | 2 | 2 | 6 | 4 | 6 | 12 |
Matrix representation of C4×C3.He3 ►in GL4(𝔽37) generated by
| 6 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 26 |
| 26 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 34 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 26 | 0 | 0 |
G:=sub<GL(4,GF(37))| [6,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,26,0,0,0,0,26,0,0,0,0,26],[26,0,0,0,0,7,0,0,0,0,7,0,0,0,0,34],[1,0,0,0,0,1,0,0,0,0,26,0,0,0,0,10],[1,0,0,0,0,0,0,26,0,1,0,0,0,0,1,0] >;
C4×C3.He3 in GAP, Magma, Sage, TeX
C_4\times C_3.{\rm He}_3 % in TeX
G:=Group("C4xC3.He3"); // GroupNames label
G:=SmallGroup(324,34);
// by ID
G=gap.SmallGroup(324,34);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,655,386,662,2170]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=d^3=1,c^3=b^-1,e^3=b,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*c*d^-1,e*d*e^-1=b^-1*d>;
// generators/relations
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