direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C4×He3⋊C3, He3⋊4C12, C12.4He3, (C3×C36)⋊3C3, (C3×C9)⋊10C12, (C4×He3)⋊2C3, C6.5(C2×He3), C3.4(C4×He3), (C3×C18).13C6, (C2×He3).8C6, (C3×C12).3C32, C32.3(C3×C12), (C3×C6).4(C3×C6), C2.(C2×He3⋊C3), (C2×He3⋊C3).4C2, SmallGroup(324,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×He3⋊C3
G = < a,b,c,d,e | a4=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ce=ec, ede-1=bcd >
Smallest permutation representation of C4×He3⋊C3
►On 108 points
Generators in S108
(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 16 60)(2 13 57)(3 14 58)(4 15 59)(5 90 35)(6 91 36)(7 92 33)(8 89 34)(9 100 40)(10 97 37)(11 98 38)(12 99 39)(17 102 76)(18 103 73)(19 104 74)(20 101 75)(21 66 49)(22 67 50)(23 68 51)(24 65 52)(25 61 107)(26 62 108)(27 63 105)(28 64 106)(29 53 80)(30 54 77)(31 55 78)(32 56 79)(41 86 82)(42 87 83)(43 88 84)(44 85 81)(45 93 71)(46 94 72)(47 95 69)(48 96 70)
(1 81 35)(2 82 36)(3 83 33)(4 84 34)(5 16 44)(6 13 41)(7 14 42)(8 15 43)(9 30 105)(10 31 106)(11 32 107)(12 29 108)(17 22 95)(18 23 96)(19 24 93)(20 21 94)(25 98 56)(26 99 53)(27 100 54)(28 97 55)(37 78 64)(38 79 61)(39 80 62)(40 77 63)(45 74 52)(46 75 49)(47 76 50)(48 73 51)(57 86 91)(58 87 92)(59 88 89)(60 85 90)(65 71 104)(66 72 101)(67 69 102)(68 70 103)
(1 70 30)(2 71 31)(3 72 32)(4 69 29)(5 48 27)(6 45 28)(7 46 25)(8 47 26)(9 35 68)(10 36 65)(11 33 66)(12 34 67)(13 74 97)(14 75 98)(15 76 99)(16 73 100)(17 80 89)(18 77 90)(19 78 91)(20 79 92)(21 61 58)(22 62 59)(23 63 60)(24 64 57)(37 86 93)(38 87 94)(39 88 95)(40 85 96)(41 52 55)(42 49 56)(43 50 53)(44 51 54)(81 103 105)(82 104 106)(83 101 107)(84 102 108)
(5 16 44)(6 13 41)(7 14 42)(8 15 43)(9 27 63)(10 28 64)(11 25 61)(12 26 62)(17 47 102)(18 48 103)(19 45 104)(20 46 101)(21 75 66)(22 76 67)(23 73 68)(24 74 65)(29 99 39)(30 100 40)(31 97 37)(32 98 38)(49 72 94)(50 69 95)(51 70 96)(52 71 93)(53 80 108)(54 77 105)(55 78 106)(56 79 107)(57 91 86)(58 92 87)(59 89 88)(60 90 85)
G:=sub<Sym(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), (1,16,60)(2,13,57)(3,14,58)(4,15,59)(5,90,35)(6,91,36)(7,92,33)(8,89,34)(9,100,40)(10,97,37)(11,98,38)(12,99,39)(17,102,76)(18,103,73)(19,104,74)(20,101,75)(21,66,49)(22,67,50)(23,68,51)(24,65,52)(25,61,107)(26,62,108)(27,63,105)(28,64,106)(29,53,80)(30,54,77)(31,55,78)(32,56,79)(41,86,82)(42,87,83)(43,88,84)(44,85,81)(45,93,71)(46,94,72)(47,95,69)(48,96,70), (1,81,35)(2,82,36)(3,83,33)(4,84,34)(5,16,44)(6,13,41)(7,14,42)(8,15,43)(9,30,105)(10,31,106)(11,32,107)(12,29,108)(17,22,95)(18,23,96)(19,24,93)(20,21,94)(25,98,56)(26,99,53)(27,100,54)(28,97,55)(37,78,64)(38,79,61)(39,80,62)(40,77,63)(45,74,52)(46,75,49)(47,76,50)(48,73,51)(57,86,91)(58,87,92)(59,88,89)(60,85,90)(65,71,104)(66,72,101)(67,69,102)(68,70,103), (1,70,30)(2,71,31)(3,72,32)(4,69,29)(5,48,27)(6,45,28)(7,46,25)(8,47,26)(9,35,68)(10,36,65)(11,33,66)(12,34,67)(13,74,97)(14,75,98)(15,76,99)(16,73,100)(17,80,89)(18,77,90)(19,78,91)(20,79,92)(21,61,58)(22,62,59)(23,63,60)(24,64,57)(37,86,93)(38,87,94)(39,88,95)(40,85,96)(41,52,55)(42,49,56)(43,50,53)(44,51,54)(81,103,105)(82,104,106)(83,101,107)(84,102,108), (5,16,44)(6,13,41)(7,14,42)(8,15,43)(9,27,63)(10,28,64)(11,25,61)(12,26,62)(17,47,102)(18,48,103)(19,45,104)(20,46,101)(21,75,66)(22,76,67)(23,73,68)(24,74,65)(29,99,39)(30,100,40)(31,97,37)(32,98,38)(49,72,94)(50,69,95)(51,70,96)(52,71,93)(53,80,108)(54,77,105)(55,78,106)(56,79,107)(57,91,86)(58,92,87)(59,89,88)(60,90,85)>;
G:=Group( (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), (1,16,60)(2,13,57)(3,14,58)(4,15,59)(5,90,35)(6,91,36)(7,92,33)(8,89,34)(9,100,40)(10,97,37)(11,98,38)(12,99,39)(17,102,76)(18,103,73)(19,104,74)(20,101,75)(21,66,49)(22,67,50)(23,68,51)(24,65,52)(25,61,107)(26,62,108)(27,63,105)(28,64,106)(29,53,80)(30,54,77)(31,55,78)(32,56,79)(41,86,82)(42,87,83)(43,88,84)(44,85,81)(45,93,71)(46,94,72)(47,95,69)(48,96,70), (1,81,35)(2,82,36)(3,83,33)(4,84,34)(5,16,44)(6,13,41)(7,14,42)(8,15,43)(9,30,105)(10,31,106)(11,32,107)(12,29,108)(17,22,95)(18,23,96)(19,24,93)(20,21,94)(25,98,56)(26,99,53)(27,100,54)(28,97,55)(37,78,64)(38,79,61)(39,80,62)(40,77,63)(45,74,52)(46,75,49)(47,76,50)(48,73,51)(57,86,91)(58,87,92)(59,88,89)(60,85,90)(65,71,104)(66,72,101)(67,69,102)(68,70,103), (1,70,30)(2,71,31)(3,72,32)(4,69,29)(5,48,27)(6,45,28)(7,46,25)(8,47,26)(9,35,68)(10,36,65)(11,33,66)(12,34,67)(13,74,97)(14,75,98)(15,76,99)(16,73,100)(17,80,89)(18,77,90)(19,78,91)(20,79,92)(21,61,58)(22,62,59)(23,63,60)(24,64,57)(37,86,93)(38,87,94)(39,88,95)(40,85,96)(41,52,55)(42,49,56)(43,50,53)(44,51,54)(81,103,105)(82,104,106)(83,101,107)(84,102,108), (5,16,44)(6,13,41)(7,14,42)(8,15,43)(9,27,63)(10,28,64)(11,25,61)(12,26,62)(17,47,102)(18,48,103)(19,45,104)(20,46,101)(21,75,66)(22,76,67)(23,73,68)(24,74,65)(29,99,39)(30,100,40)(31,97,37)(32,98,38)(49,72,94)(50,69,95)(51,70,96)(52,71,93)(53,80,108)(54,77,105)(55,78,106)(56,79,107)(57,91,86)(58,92,87)(59,89,88)(60,90,85) );
G=PermutationGroup([[(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)], [(1,16,60),(2,13,57),(3,14,58),(4,15,59),(5,90,35),(6,91,36),(7,92,33),(8,89,34),(9,100,40),(10,97,37),(11,98,38),(12,99,39),(17,102,76),(18,103,73),(19,104,74),(20,101,75),(21,66,49),(22,67,50),(23,68,51),(24,65,52),(25,61,107),(26,62,108),(27,63,105),(28,64,106),(29,53,80),(30,54,77),(31,55,78),(32,56,79),(41,86,82),(42,87,83),(43,88,84),(44,85,81),(45,93,71),(46,94,72),(47,95,69),(48,96,70)], [(1,81,35),(2,82,36),(3,83,33),(4,84,34),(5,16,44),(6,13,41),(7,14,42),(8,15,43),(9,30,105),(10,31,106),(11,32,107),(12,29,108),(17,22,95),(18,23,96),(19,24,93),(20,21,94),(25,98,56),(26,99,53),(27,100,54),(28,97,55),(37,78,64),(38,79,61),(39,80,62),(40,77,63),(45,74,52),(46,75,49),(47,76,50),(48,73,51),(57,86,91),(58,87,92),(59,88,89),(60,85,90),(65,71,104),(66,72,101),(67,69,102),(68,70,103)], [(1,70,30),(2,71,31),(3,72,32),(4,69,29),(5,48,27),(6,45,28),(7,46,25),(8,47,26),(9,35,68),(10,36,65),(11,33,66),(12,34,67),(13,74,97),(14,75,98),(15,76,99),(16,73,100),(17,80,89),(18,77,90),(19,78,91),(20,79,92),(21,61,58),(22,62,59),(23,63,60),(24,64,57),(37,86,93),(38,87,94),(39,88,95),(40,85,96),(41,52,55),(42,49,56),(43,50,53),(44,51,54),(81,103,105),(82,104,106),(83,101,107),(84,102,108)], [(5,16,44),(6,13,41),(7,14,42),(8,15,43),(9,27,63),(10,28,64),(11,25,61),(12,26,62),(17,47,102),(18,48,103),(19,45,104),(20,46,101),(21,75,66),(22,76,67),(23,73,68),(24,74,65),(29,99,39),(30,100,40),(31,97,37),(32,98,38),(49,72,94),(50,69,95),(51,70,96),(52,71,93),(53,80,108),(54,77,105),(55,78,106),(56,79,107),(57,91,86),(58,92,87),(59,89,88),(60,90,85)]])
68 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 4A | 4B | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12T | 18A | ··· | 18F | 36A | ··· | 36L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 9 | ··· | 9 | 1 | 1 | 1 | 1 | 3 | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 3 | ··· | 3 |
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 | He3⋊C3 | C4×He3 | C2×He3⋊C3 | C4×He3⋊C3 |
| kernel | C4×He3⋊C3 | C2×He3⋊C3 | C3×C36 | C4×He3 | He3⋊C3 | C3×C18 | C2×He3 | C3×C9 | He3 | 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×He3⋊C3 ►in GL3(𝔽37) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 16 | 9 | 9 |
| 12 | 9 | 12 |
| 16 | 16 | 12 |
| 1 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 10 |
G:=sub<GL(3,GF(37))| [6,0,0,0,6,0,0,0,6],[0,0,1,1,0,0,0,1,0],[10,0,0,0,10,0,0,0,10],[16,12,16,9,9,16,9,12,12],[1,0,0,0,26,0,0,0,10] >;
C4×He3⋊C3 in GAP, Magma, Sage, TeX
C_4\times {\rm He}_3\rtimes C_3 % in TeX
G:=Group("C4xHe3:C3"); // GroupNames label
G:=SmallGroup(324,33);
// by ID
G=gap.SmallGroup(324,33);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,662,2170]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=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,d*b*d^-1=e*b*e^-1=b*c^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
// generators/relations
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