direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C34, C23⋊C51, C22⋊C102, (C22×C34)⋊C3, (C2×C34)⋊2C6, SmallGroup(408,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C34 |
Generators and relations for A4×C34
G = < a,b,c,d | a34=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C34
►On 102 points
Generators in S102
(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)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)
(1 93 39)(2 94 40)(3 95 41)(4 96 42)(5 97 43)(6 98 44)(7 99 45)(8 100 46)(9 101 47)(10 102 48)(11 69 49)(12 70 50)(13 71 51)(14 72 52)(15 73 53)(16 74 54)(17 75 55)(18 76 56)(19 77 57)(20 78 58)(21 79 59)(22 80 60)(23 81 61)(24 82 62)(25 83 63)(26 84 64)(27 85 65)(28 86 66)(29 87 67)(30 88 68)(31 89 35)(32 90 36)(33 91 37)(34 92 38)
G:=sub<Sym(102)| (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), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102), (1,93,39)(2,94,40)(3,95,41)(4,96,42)(5,97,43)(6,98,44)(7,99,45)(8,100,46)(9,101,47)(10,102,48)(11,69,49)(12,70,50)(13,71,51)(14,72,52)(15,73,53)(16,74,54)(17,75,55)(18,76,56)(19,77,57)(20,78,58)(21,79,59)(22,80,60)(23,81,61)(24,82,62)(25,83,63)(26,84,64)(27,85,65)(28,86,66)(29,87,67)(30,88,68)(31,89,35)(32,90,36)(33,91,37)(34,92,38)>;
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), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102), (1,93,39)(2,94,40)(3,95,41)(4,96,42)(5,97,43)(6,98,44)(7,99,45)(8,100,46)(9,101,47)(10,102,48)(11,69,49)(12,70,50)(13,71,51)(14,72,52)(15,73,53)(16,74,54)(17,75,55)(18,76,56)(19,77,57)(20,78,58)(21,79,59)(22,80,60)(23,81,61)(24,82,62)(25,83,63)(26,84,64)(27,85,65)(28,86,66)(29,87,67)(30,88,68)(31,89,35)(32,90,36)(33,91,37)(34,92,38) );
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)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,68)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(85,102)], [(1,93,39),(2,94,40),(3,95,41),(4,96,42),(5,97,43),(6,98,44),(7,99,45),(8,100,46),(9,101,47),(10,102,48),(11,69,49),(12,70,50),(13,71,51),(14,72,52),(15,73,53),(16,74,54),(17,75,55),(18,76,56),(19,77,57),(20,78,58),(21,79,59),(22,80,60),(23,81,61),(24,82,62),(25,83,63),(26,84,64),(27,85,65),(28,86,66),(29,87,67),(30,88,68),(31,89,35),(32,90,36),(33,91,37),(34,92,38)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 51A | ··· | 51AF | 102A | ··· | 102AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C17 | C34 | C51 | C102 | A4 | C2×A4 | A4×C17 | A4×C34 |
| kernel | A4×C34 | A4×C17 | C22×C34 | C2×C34 | C2×A4 | A4 | C23 | C22 | C34 | C17 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 16 | 16 | 32 | 32 | 1 | 1 | 16 | 16 |
Matrix representation of A4×C34 ►in GL3(𝔽103) generated by
| 94 | 0 | 0 |
| 0 | 94 | 0 |
| 0 | 0 | 94 |
| 102 | 0 | 0 |
| 0 | 102 | 0 |
| 20 | 0 | 1 |
| 102 | 0 | 0 |
| 3 | 1 | 0 |
| 0 | 0 | 102 |
| 100 | 101 | 0 |
| 66 | 3 | 1 |
| 81 | 20 | 0 |
G:=sub<GL(3,GF(103))| [94,0,0,0,94,0,0,0,94],[102,0,20,0,102,0,0,0,1],[102,3,0,0,1,0,0,0,102],[100,66,81,101,3,20,0,1,0] >;
A4×C34 in GAP, Magma, Sage, TeX
A_4\times C_{34} % in TeX
G:=Group("A4xC34"); // GroupNames label
G:=SmallGroup(408,42);
// by ID
G=gap.SmallGroup(408,42);
# by ID
G:=PCGroup([5,-2,-3,-17,-2,2,2048,3834]);
// Polycyclic
G:=Group<a,b,c,d|a^34=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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