direct product, metabelian, nilpotent (class 3), monomial
Aliases: C22×He3⋊C3, C32.3C62, C62.21C32, (C6×C18)⋊6C3, (C3×C18)⋊10C6, (C2×He3)⋊3C6, He3⋊4(C2×C6), C6.10(C2×He3), (C2×C6).16He3, (C22×He3)⋊4C3, C3.4(C22×He3), (C3×C9)⋊12(C2×C6), (C3×C6).8(C3×C6), SmallGroup(324,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×He3⋊C3
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=fcf-1=cd-1, de=ed, df=fd, fef-1=cde >
Subgroups: 250 in 80 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C3, C3, C22, C6, C6, C9, C32, C32, C2×C6, C2×C6, C18, C3×C6, C3×C6, C3×C9, He3, C2×C18, C62, C62, C3×C18, C2×He3, He3⋊C3, C6×C18, C22×He3, C2×He3⋊C3, C22×He3⋊C3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, He3, C62, C2×He3, He3⋊C3, C22×He3, C2×He3⋊C3, C22×He3⋊C3
►On 108 points
Generators in S108
(1 61)(2 62)(3 63)(4 57)(5 55)(6 56)(7 60)(8 58)(9 59)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(1 34)(2 35)(3 36)(4 84)(5 82)(6 83)(7 87)(8 85)(9 86)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)
(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 14 10)(2 15 11)(3 13 12)(4 108 8)(5 106 9)(6 107 7)(16 23 19)(17 24 20)(18 22 21)(25 32 28)(26 33 29)(27 31 30)(34 41 37)(35 42 38)(36 40 39)(43 50 46)(44 51 47)(45 49 48)(52 59 55)(53 60 56)(54 58 57)(61 68 64)(62 69 65)(63 67 66)(70 77 73)(71 78 74)(72 76 75)(79 86 82)(80 87 83)(81 85 84)(88 95 91)(89 96 92)(90 94 93)(97 104 100)(98 105 101)(99 103 102)
(2 15 11)(3 12 13)(4 9 7)(5 6 108)(8 106 107)(16 18 20)(17 23 22)(19 21 24)(25 26 31)(27 28 29)(30 32 33)(35 42 38)(36 39 40)(43 45 47)(44 50 49)(46 48 51)(52 53 58)(54 55 56)(57 59 60)(62 69 65)(63 66 67)(70 72 74)(71 77 76)(73 75 78)(79 80 85)(81 82 83)(84 86 87)(89 96 92)(90 93 94)(97 99 101)(98 104 103)(100 102 105)
(1 28 20)(2 26 22)(3 31 16)(4 104 94)(5 101 88)(6 99 92)(7 102 96)(8 97 90)(9 105 91)(10 32 24)(11 29 18)(12 27 19)(13 30 23)(14 25 17)(15 33 21)(34 55 47)(35 53 49)(36 58 43)(37 59 51)(38 56 45)(39 54 46)(40 57 50)(41 52 44)(42 60 48)(61 82 74)(62 80 76)(63 85 70)(64 86 78)(65 83 72)(66 81 73)(67 84 77)(68 79 71)(69 87 75)(89 107 103)(93 108 100)(95 106 98)
G:=sub<Sym(108)| (1,61)(2,62)(3,63)(4,57)(5,55)(6,56)(7,60)(8,58)(9,59)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,34)(2,35)(3,36)(4,84)(5,82)(6,83)(7,87)(8,85)(9,86)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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), (1,14,10)(2,15,11)(3,13,12)(4,108,8)(5,106,9)(6,107,7)(16,23,19)(17,24,20)(18,22,21)(25,32,28)(26,33,29)(27,31,30)(34,41,37)(35,42,38)(36,40,39)(43,50,46)(44,51,47)(45,49,48)(52,59,55)(53,60,56)(54,58,57)(61,68,64)(62,69,65)(63,67,66)(70,77,73)(71,78,74)(72,76,75)(79,86,82)(80,87,83)(81,85,84)(88,95,91)(89,96,92)(90,94,93)(97,104,100)(98,105,101)(99,103,102), (2,15,11)(3,12,13)(4,9,7)(5,6,108)(8,106,107)(16,18,20)(17,23,22)(19,21,24)(25,26,31)(27,28,29)(30,32,33)(35,42,38)(36,39,40)(43,45,47)(44,50,49)(46,48,51)(52,53,58)(54,55,56)(57,59,60)(62,69,65)(63,66,67)(70,72,74)(71,77,76)(73,75,78)(79,80,85)(81,82,83)(84,86,87)(89,96,92)(90,93,94)(97,99,101)(98,104,103)(100,102,105), (1,28,20)(2,26,22)(3,31,16)(4,104,94)(5,101,88)(6,99,92)(7,102,96)(8,97,90)(9,105,91)(10,32,24)(11,29,18)(12,27,19)(13,30,23)(14,25,17)(15,33,21)(34,55,47)(35,53,49)(36,58,43)(37,59,51)(38,56,45)(39,54,46)(40,57,50)(41,52,44)(42,60,48)(61,82,74)(62,80,76)(63,85,70)(64,86,78)(65,83,72)(66,81,73)(67,84,77)(68,79,71)(69,87,75)(89,107,103)(93,108,100)(95,106,98)>;
G:=Group( (1,61)(2,62)(3,63)(4,57)(5,55)(6,56)(7,60)(8,58)(9,59)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,34)(2,35)(3,36)(4,84)(5,82)(6,83)(7,87)(8,85)(9,86)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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), (1,14,10)(2,15,11)(3,13,12)(4,108,8)(5,106,9)(6,107,7)(16,23,19)(17,24,20)(18,22,21)(25,32,28)(26,33,29)(27,31,30)(34,41,37)(35,42,38)(36,40,39)(43,50,46)(44,51,47)(45,49,48)(52,59,55)(53,60,56)(54,58,57)(61,68,64)(62,69,65)(63,67,66)(70,77,73)(71,78,74)(72,76,75)(79,86,82)(80,87,83)(81,85,84)(88,95,91)(89,96,92)(90,94,93)(97,104,100)(98,105,101)(99,103,102), (2,15,11)(3,12,13)(4,9,7)(5,6,108)(8,106,107)(16,18,20)(17,23,22)(19,21,24)(25,26,31)(27,28,29)(30,32,33)(35,42,38)(36,39,40)(43,45,47)(44,50,49)(46,48,51)(52,53,58)(54,55,56)(57,59,60)(62,69,65)(63,66,67)(70,72,74)(71,77,76)(73,75,78)(79,80,85)(81,82,83)(84,86,87)(89,96,92)(90,93,94)(97,99,101)(98,104,103)(100,102,105), (1,28,20)(2,26,22)(3,31,16)(4,104,94)(5,101,88)(6,99,92)(7,102,96)(8,97,90)(9,105,91)(10,32,24)(11,29,18)(12,27,19)(13,30,23)(14,25,17)(15,33,21)(34,55,47)(35,53,49)(36,58,43)(37,59,51)(38,56,45)(39,54,46)(40,57,50)(41,52,44)(42,60,48)(61,82,74)(62,80,76)(63,85,70)(64,86,78)(65,83,72)(66,81,73)(67,84,77)(68,79,71)(69,87,75)(89,107,103)(93,108,100)(95,106,98) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,57),(5,55),(6,56),(7,60),(8,58),(9,59),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108)], [(1,34),(2,35),(3,36),(4,84),(5,82),(6,83),(7,87),(8,85),(9,86),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108)], [(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)], [(1,14,10),(2,15,11),(3,13,12),(4,108,8),(5,106,9),(6,107,7),(16,23,19),(17,24,20),(18,22,21),(25,32,28),(26,33,29),(27,31,30),(34,41,37),(35,42,38),(36,40,39),(43,50,46),(44,51,47),(45,49,48),(52,59,55),(53,60,56),(54,58,57),(61,68,64),(62,69,65),(63,67,66),(70,77,73),(71,78,74),(72,76,75),(79,86,82),(80,87,83),(81,85,84),(88,95,91),(89,96,92),(90,94,93),(97,104,100),(98,105,101),(99,103,102)], [(2,15,11),(3,12,13),(4,9,7),(5,6,108),(8,106,107),(16,18,20),(17,23,22),(19,21,24),(25,26,31),(27,28,29),(30,32,33),(35,42,38),(36,39,40),(43,45,47),(44,50,49),(46,48,51),(52,53,58),(54,55,56),(57,59,60),(62,69,65),(63,66,67),(70,72,74),(71,77,76),(73,75,78),(79,80,85),(81,82,83),(84,86,87),(89,96,92),(90,93,94),(97,99,101),(98,104,103),(100,102,105)], [(1,28,20),(2,26,22),(3,31,16),(4,104,94),(5,101,88),(6,99,92),(7,102,96),(8,97,90),(9,105,91),(10,32,24),(11,29,18),(12,27,19),(13,30,23),(14,25,17),(15,33,21),(34,55,47),(35,53,49),(36,58,43),(37,59,51),(38,56,45),(39,54,46),(40,57,50),(41,52,44),(42,60,48),(61,82,74),(62,80,76),(63,85,70),(64,86,78),(65,83,72),(66,81,73),(67,84,77),(68,79,71),(69,87,75),(89,107,103),(93,108,100),(95,106,98)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | ··· | 6F | 6G | ··· | 6L | 6M | ··· | 6AD | 9A | ··· | 9F | 18A | ··· | 18R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 9 | ··· | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 3 | ··· | 3 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | He3 | C2×He3 | He3⋊C3 | C2×He3⋊C3 |
| kernel | C22×He3⋊C3 | C2×He3⋊C3 | C6×C18 | C22×He3 | C3×C18 | C2×He3 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 2 | 6 | 6 | 18 |
Matrix representation of C22×He3⋊C3 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 10 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 11 |
| 0 | 0 | 11 | 12 |
| 0 | 0 | 0 | 7 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 2 | 16 |
| 0 | 17 | 14 | 14 |
| 0 | 5 | 5 | 5 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,8,10,0,1,0,0,0,0,0,7],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[7,0,0,0,0,1,0,0,0,0,11,0,0,11,12,7],[7,0,0,0,0,0,17,5,0,2,14,5,0,16,14,5] >;
C22×He3⋊C3 in GAP, Magma, Sage, TeX
C_2^2\times {\rm He}_3\rtimes C_3 % in TeX
G:=Group("C2^2xHe3:C3"); // GroupNames label
G:=SmallGroup(324,88);
// by ID
G=gap.SmallGroup(324,88);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,453,1096]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^3=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,e*c*e^-1=f*c*f^-1=c*d^-1,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e>;
// generators/relations