direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×C4≀C2, D4⋊2C44, Q8⋊2C44, C42⋊3C22, C44.66D4, M4(2)⋊4C22, (C4×C44)⋊10C2, (D4×C11)⋊5C4, C4.3(C2×C44), (Q8×C11)⋊5C4, C44.30(C2×C4), C4○D4.1C22, C4.17(D4×C11), (C2×C22).22D4, C22.3(D4×C11), C22.26(C22⋊C4), (C11×M4(2))⋊10C2, (C2×C44).116C22, (C2×C4).19(C2×C22), (C11×C4○D4).4C2, C2.8(C11×C22⋊C4), SmallGroup(352,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C4≀C2
G = < a,b,c,d | a11=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Smallest permutation representation of C11×C4≀C2
►On 88 points
Generators in S88
(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)
(1 59 46 75)(2 60 47 76)(3 61 48 77)(4 62 49 67)(5 63 50 68)(6 64 51 69)(7 65 52 70)(8 66 53 71)(9 56 54 72)(10 57 55 73)(11 58 45 74)(12 88 25 38)(13 78 26 39)(14 79 27 40)(15 80 28 41)(16 81 29 42)(17 82 30 43)(18 83 31 44)(19 84 32 34)(20 85 33 35)(21 86 23 36)(22 87 24 37)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 73)(13 74)(14 75)(15 76)(16 77)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 66)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 45)(12 88 25 38)(13 78 26 39)(14 79 27 40)(15 80 28 41)(16 81 29 42)(17 82 30 43)(18 83 31 44)(19 84 32 34)(20 85 33 35)(21 86 23 36)(22 87 24 37)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 67)(63 68)(64 69)(65 70)(66 71)
G:=sub<Sym(88)| (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), (1,59,46,75)(2,60,47,76)(3,61,48,77)(4,62,49,67)(5,63,50,68)(6,64,51,69)(7,65,52,70)(8,66,53,71)(9,56,54,72)(10,57,55,73)(11,58,45,74)(12,88,25,38)(13,78,26,39)(14,79,27,40)(15,80,28,41)(16,81,29,42)(17,82,30,43)(18,83,31,44)(19,84,32,34)(20,85,33,35)(21,86,23,36)(22,87,24,37), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,73)(13,74)(14,75)(15,76)(16,77)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,66)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,88,25,38)(13,78,26,39)(14,79,27,40)(15,80,28,41)(16,81,29,42)(17,82,30,43)(18,83,31,44)(19,84,32,34)(20,85,33,35)(21,86,23,36)(22,87,24,37)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,67)(63,68)(64,69)(65,70)(66,71)>;
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), (1,59,46,75)(2,60,47,76)(3,61,48,77)(4,62,49,67)(5,63,50,68)(6,64,51,69)(7,65,52,70)(8,66,53,71)(9,56,54,72)(10,57,55,73)(11,58,45,74)(12,88,25,38)(13,78,26,39)(14,79,27,40)(15,80,28,41)(16,81,29,42)(17,82,30,43)(18,83,31,44)(19,84,32,34)(20,85,33,35)(21,86,23,36)(22,87,24,37), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,73)(13,74)(14,75)(15,76)(16,77)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,66)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,88,25,38)(13,78,26,39)(14,79,27,40)(15,80,28,41)(16,81,29,42)(17,82,30,43)(18,83,31,44)(19,84,32,34)(20,85,33,35)(21,86,23,36)(22,87,24,37)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,67)(63,68)(64,69)(65,70)(66,71) );
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)], [(1,59,46,75),(2,60,47,76),(3,61,48,77),(4,62,49,67),(5,63,50,68),(6,64,51,69),(7,65,52,70),(8,66,53,71),(9,56,54,72),(10,57,55,73),(11,58,45,74),(12,88,25,38),(13,78,26,39),(14,79,27,40),(15,80,28,41),(16,81,29,42),(17,82,30,43),(18,83,31,44),(19,84,32,34),(20,85,33,35),(21,86,23,36),(22,87,24,37)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,73),(13,74),(14,75),(15,76),(16,77),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,66),(24,56),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,65),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,45),(12,88,25,38),(13,78,26,39),(14,79,27,40),(15,80,28,41),(16,81,29,42),(17,82,30,43),(18,83,31,44),(19,84,32,34),(20,85,33,35),(21,86,23,36),(22,87,24,37),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,67),(63,68),(64,69),(65,70),(66,71)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 8A | 8B | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22T | 22U | ··· | 22AD | 44A | ··· | 44T | 44U | ··· | 44BR | 44BS | ··· | 44CB | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C11 | C22 | C22 | C22 | C44 | C44 | D4 | D4 | C4≀C2 | D4×C11 | D4×C11 | C11×C4≀C2 |
| kernel | C11×C4≀C2 | C4×C44 | C11×M4(2) | C11×C4○D4 | D4×C11 | Q8×C11 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C44 | C2×C22 | C11 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 1 | 1 | 4 | 10 | 10 | 40 |
Matrix representation of C11×C4≀C2 ►in GL2(𝔽89) generated by
| 45 | 0 |
| 0 | 45 |
| 34 | 0 |
| 0 | 55 |
| 0 | 55 |
| 34 | 0 |
| 88 | 0 |
| 0 | 55 |
G:=sub<GL(2,GF(89))| [45,0,0,45],[34,0,0,55],[0,34,55,0],[88,0,0,55] >;
C11×C4≀C2 in GAP, Magma, Sage, TeX
C_{11}\times C_4\wr C_2 % in TeX
G:=Group("C11xC4wrC2"); // GroupNames label
G:=SmallGroup(352,53);
// by ID
G=gap.SmallGroup(352,53);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,5283,2649,117,88]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations
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