direct product, metabelian, nilpotent (class 2), monomial
Aliases: C2×C6×He3, C32⋊3C62, C62⋊3C32, (C32×C6)⋊6C6, (C3×C62)⋊4C3, C33⋊10(C2×C6), C3.1(C3×C62), C6.5(C32×C6), (C2×C6).11C33, (C3×C6)⋊2(C3×C6), SmallGroup(324,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6×He3
G = < a,b,c,d,e | a2=b6=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 520 in 280 conjugacy classes, 160 normal (8 characteristic)
C1, C2, C3, C3, C3, C22, C6, C6, C32, C32, C32, C2×C6, C2×C6, C2×C6, C3×C6, C3×C6, He3, C33, C62, C62, C62, C2×He3, C32×C6, C3×He3, C22×He3, C3×C62, C6×He3, C2×C6×He3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, He3, C33, C62, C2×He3, C32×C6, C3×He3, C22×He3, C3×C62, C6×He3, C2×C6×He3
►On 108 points
Generators in S108
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 92)(14 93)(15 94)(16 95)(17 96)(18 91)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 87)(26 88)(27 89)(28 90)(29 85)(30 86)(31 62)(32 63)(33 64)(34 65)(35 66)(36 61)(37 102)(38 97)(39 98)(40 99)(41 100)(42 101)(43 83)(44 84)(45 79)(46 80)(47 81)(48 82)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(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)
(1 19 53)(2 20 54)(3 21 49)(4 22 50)(5 23 51)(6 24 52)(7 101 94)(8 102 95)(9 97 96)(10 98 91)(11 99 92)(12 100 93)(13 59 40)(14 60 41)(15 55 42)(16 56 37)(17 57 38)(18 58 39)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(43 45 47)(44 46 48)(61 63 65)(62 64 66)(67 73 107)(68 74 108)(69 75 103)(70 76 104)(71 77 105)(72 78 106)(79 81 83)(80 82 84)(85 87 89)(86 88 90)
(1 23 49)(2 24 50)(3 19 51)(4 20 52)(5 21 53)(6 22 54)(7 96 99)(8 91 100)(9 92 101)(10 93 102)(11 94 97)(12 95 98)(13 42 57)(14 37 58)(15 38 59)(16 39 60)(17 40 55)(18 41 56)(25 34 83)(26 35 84)(27 36 79)(28 31 80)(29 32 81)(30 33 82)(43 87 65)(44 88 66)(45 89 61)(46 90 62)(47 85 63)(48 86 64)(67 77 103)(68 78 104)(69 73 105)(70 74 106)(71 75 107)(72 76 108)
(1 89 38)(2 90 39)(3 85 40)(4 86 41)(5 87 42)(6 88 37)(7 73 32)(8 74 33)(9 75 34)(10 76 35)(11 77 36)(12 78 31)(13 53 43)(14 54 44)(15 49 45)(16 50 46)(17 51 47)(18 52 48)(19 63 55)(20 64 56)(21 65 57)(22 66 58)(23 61 59)(24 62 60)(25 101 71)(26 102 72)(27 97 67)(28 98 68)(29 99 69)(30 100 70)(79 94 103)(80 95 104)(81 96 105)(82 91 106)(83 92 107)(84 93 108)
G:=sub<Sym(108)| (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,92)(14,93)(15,94)(16,95)(17,96)(18,91)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,87)(26,88)(27,89)(28,90)(29,85)(30,86)(31,62)(32,63)(33,64)(34,65)(35,66)(36,61)(37,102)(38,97)(39,98)(40,99)(41,100)(42,101)(43,83)(44,84)(45,79)(46,80)(47,81)(48,82)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (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), (1,19,53)(2,20,54)(3,21,49)(4,22,50)(5,23,51)(6,24,52)(7,101,94)(8,102,95)(9,97,96)(10,98,91)(11,99,92)(12,100,93)(13,59,40)(14,60,41)(15,55,42)(16,56,37)(17,57,38)(18,58,39)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(43,45,47)(44,46,48)(61,63,65)(62,64,66)(67,73,107)(68,74,108)(69,75,103)(70,76,104)(71,77,105)(72,78,106)(79,81,83)(80,82,84)(85,87,89)(86,88,90), (1,23,49)(2,24,50)(3,19,51)(4,20,52)(5,21,53)(6,22,54)(7,96,99)(8,91,100)(9,92,101)(10,93,102)(11,94,97)(12,95,98)(13,42,57)(14,37,58)(15,38,59)(16,39,60)(17,40,55)(18,41,56)(25,34,83)(26,35,84)(27,36,79)(28,31,80)(29,32,81)(30,33,82)(43,87,65)(44,88,66)(45,89,61)(46,90,62)(47,85,63)(48,86,64)(67,77,103)(68,78,104)(69,73,105)(70,74,106)(71,75,107)(72,76,108), (1,89,38)(2,90,39)(3,85,40)(4,86,41)(5,87,42)(6,88,37)(7,73,32)(8,74,33)(9,75,34)(10,76,35)(11,77,36)(12,78,31)(13,53,43)(14,54,44)(15,49,45)(16,50,46)(17,51,47)(18,52,48)(19,63,55)(20,64,56)(21,65,57)(22,66,58)(23,61,59)(24,62,60)(25,101,71)(26,102,72)(27,97,67)(28,98,68)(29,99,69)(30,100,70)(79,94,103)(80,95,104)(81,96,105)(82,91,106)(83,92,107)(84,93,108)>;
G:=Group( (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,92)(14,93)(15,94)(16,95)(17,96)(18,91)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,87)(26,88)(27,89)(28,90)(29,85)(30,86)(31,62)(32,63)(33,64)(34,65)(35,66)(36,61)(37,102)(38,97)(39,98)(40,99)(41,100)(42,101)(43,83)(44,84)(45,79)(46,80)(47,81)(48,82)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (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), (1,19,53)(2,20,54)(3,21,49)(4,22,50)(5,23,51)(6,24,52)(7,101,94)(8,102,95)(9,97,96)(10,98,91)(11,99,92)(12,100,93)(13,59,40)(14,60,41)(15,55,42)(16,56,37)(17,57,38)(18,58,39)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(43,45,47)(44,46,48)(61,63,65)(62,64,66)(67,73,107)(68,74,108)(69,75,103)(70,76,104)(71,77,105)(72,78,106)(79,81,83)(80,82,84)(85,87,89)(86,88,90), (1,23,49)(2,24,50)(3,19,51)(4,20,52)(5,21,53)(6,22,54)(7,96,99)(8,91,100)(9,92,101)(10,93,102)(11,94,97)(12,95,98)(13,42,57)(14,37,58)(15,38,59)(16,39,60)(17,40,55)(18,41,56)(25,34,83)(26,35,84)(27,36,79)(28,31,80)(29,32,81)(30,33,82)(43,87,65)(44,88,66)(45,89,61)(46,90,62)(47,85,63)(48,86,64)(67,77,103)(68,78,104)(69,73,105)(70,74,106)(71,75,107)(72,76,108), (1,89,38)(2,90,39)(3,85,40)(4,86,41)(5,87,42)(6,88,37)(7,73,32)(8,74,33)(9,75,34)(10,76,35)(11,77,36)(12,78,31)(13,53,43)(14,54,44)(15,49,45)(16,50,46)(17,51,47)(18,52,48)(19,63,55)(20,64,56)(21,65,57)(22,66,58)(23,61,59)(24,62,60)(25,101,71)(26,102,72)(27,97,67)(28,98,68)(29,99,69)(30,100,70)(79,94,103)(80,95,104)(81,96,105)(82,91,106)(83,92,107)(84,93,108) );
G=PermutationGroup([[(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,92),(14,93),(15,94),(16,95),(17,96),(18,91),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,87),(26,88),(27,89),(28,90),(29,85),(30,86),(31,62),(32,63),(33,64),(34,65),(35,66),(36,61),(37,102),(38,97),(39,98),(40,99),(41,100),(42,101),(43,83),(44,84),(45,79),(46,80),(47,81),(48,82),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108)], [(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)], [(1,19,53),(2,20,54),(3,21,49),(4,22,50),(5,23,51),(6,24,52),(7,101,94),(8,102,95),(9,97,96),(10,98,91),(11,99,92),(12,100,93),(13,59,40),(14,60,41),(15,55,42),(16,56,37),(17,57,38),(18,58,39),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(43,45,47),(44,46,48),(61,63,65),(62,64,66),(67,73,107),(68,74,108),(69,75,103),(70,76,104),(71,77,105),(72,78,106),(79,81,83),(80,82,84),(85,87,89),(86,88,90)], [(1,23,49),(2,24,50),(3,19,51),(4,20,52),(5,21,53),(6,22,54),(7,96,99),(8,91,100),(9,92,101),(10,93,102),(11,94,97),(12,95,98),(13,42,57),(14,37,58),(15,38,59),(16,39,60),(17,40,55),(18,41,56),(25,34,83),(26,35,84),(27,36,79),(28,31,80),(29,32,81),(30,33,82),(43,87,65),(44,88,66),(45,89,61),(46,90,62),(47,85,63),(48,86,64),(67,77,103),(68,78,104),(69,73,105),(70,74,106),(71,75,107),(72,76,108)], [(1,89,38),(2,90,39),(3,85,40),(4,86,41),(5,87,42),(6,88,37),(7,73,32),(8,74,33),(9,75,34),(10,76,35),(11,77,36),(12,78,31),(13,53,43),(14,54,44),(15,49,45),(16,50,46),(17,51,47),(18,52,48),(19,63,55),(20,64,56),(21,65,57),(22,66,58),(23,61,59),(24,62,60),(25,101,71),(26,102,72),(27,97,67),(28,98,68),(29,99,69),(30,100,70),(79,94,103),(80,95,104),(81,96,105),(82,91,106),(83,92,107),(84,93,108)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3AF | 6A | ··· | 6X | 6Y | ··· | 6CR |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | He3 | C2×He3 |
| kernel | C2×C6×He3 | C6×He3 | C22×He3 | C3×C62 | C2×He3 | C32×C6 | C2×C6 | C6 |
| # reps | 1 | 3 | 18 | 8 | 54 | 24 | 6 | 18 |
Matrix representation of C2×C6×He3 ►in GL4(𝔽7) generated by
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[2,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,2],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C2×C6×He3 in GAP, Magma, Sage, TeX
C_2\times C_6\times {\rm He}_3 % in TeX
G:=Group("C2xC6xHe3"); // GroupNames label
G:=SmallGroup(324,152);
// by ID
G=gap.SmallGroup(324,152);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,735]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^3=d^3=e^3=1,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=c*d^-1,d*e=e*d>;
// generators/relations