direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C22≀C2, C24⋊1C22, (C2×C22)⋊7D4, (C2×D4)⋊1C22, C2.4(D4×C22), (D4×C22)⋊10C2, C22⋊C4⋊2C22, (C2×C44)⋊8C22, (C23×C22)⋊1C2, C23⋊1(C2×C22), C22.67(C2×D4), C22⋊2(D4×C11), (C2×C22).75C23, (C22×C22)⋊1C22, C22.10(C22×C22), (C2×C4)⋊1(C2×C22), (C11×C22⋊C4)⋊10C2, SmallGroup(352,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C22≀C2
G = < a,b,c,d,e,f | a11=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C23, C11, C22⋊C4, C2×D4, C24, C22, C22, C22≀C2, C44, C2×C22, C2×C22, C2×C22, C2×C44, D4×C11, C22×C22, C22×C22, C22×C22, C11×C22⋊C4, D4×C22, C23×C22, C11×C22≀C2
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C22, C22≀C2, C2×C22, D4×C11, C22×C22, D4×C22, C11×C22≀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 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 45)(12 42)(13 43)(14 44)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 78)(30 79)(31 80)(32 81)(33 82)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 67)
(1 66)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 78)(21 79)(22 80)(23 44)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(45 77)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 78)(21 79)(22 80)(23 44)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(45 65)(46 66)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 45)(12 32)(13 33)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(34 84)(35 85)(36 86)(37 87)(38 88)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 67)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 45)(30 46)(31 47)(32 48)(33 49)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 56)(42 57)(43 58)(44 59)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 78)
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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,42)(13,43)(14,44)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,78)(30,79)(31,80)(32,81)(33,82)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,67), (1,66)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,78)(21,79)(22,80)(23,44)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(45,77)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,78)(21,79)(22,80)(23,44)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(45,65)(46,66)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,32)(13,33)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(34,84)(35,85)(36,86)(37,87)(38,88)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,67), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,45)(30,46)(31,47)(32,48)(33,49)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,56)(42,57)(43,58)(44,59)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,78)>;
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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,42)(13,43)(14,44)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,78)(30,79)(31,80)(32,81)(33,82)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,67), (1,66)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,78)(21,79)(22,80)(23,44)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(45,77)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,78)(21,79)(22,80)(23,44)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(45,65)(46,66)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,32)(13,33)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(34,84)(35,85)(36,86)(37,87)(38,88)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,67), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,45)(30,46)(31,47)(32,48)(33,49)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,56)(42,57)(43,58)(44,59)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,78) );
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)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,45),(12,42),(13,43),(14,44),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,78),(30,79),(31,80),(32,81),(33,82),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,67)], [(1,66),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,78),(21,79),(22,80),(23,44),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(45,77),(46,67),(47,68),(48,69),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,78),(21,79),(22,80),(23,44),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(45,65),(46,66),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,45),(12,32),(13,33),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(34,84),(35,85),(36,86),(37,87),(38,88),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,67)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,45),(30,46),(31,47),(32,48),(33,49),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,56),(42,57),(43,58),(44,59),(67,79),(68,80),(69,81),(70,82),(71,83),(72,84),(73,85),(74,86),(75,87),(76,88),(77,78)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 4A | 4B | 4C | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22CL | 22CM | ··· | 22CV | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | D4 | D4×C11 |
| kernel | C11×C22≀C2 | C11×C22⋊C4 | D4×C22 | C23×C22 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C22 | C22 |
| # reps | 1 | 3 | 3 | 1 | 10 | 30 | 30 | 10 | 6 | 60 |
Matrix representation of C11×C22≀C2 ►in GL4(𝔽89) generated by
| 67 | 0 | 0 | 0 |
| 0 | 67 | 0 | 0 |
| 0 | 0 | 78 | 0 |
| 0 | 0 | 0 | 78 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 1 | 1 |
| 88 | 0 | 0 | 0 |
| 23 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 88 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 66 | 87 | 0 | 0 |
| 86 | 23 | 0 | 0 |
| 0 | 0 | 88 | 87 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(89))| [67,0,0,0,0,67,0,0,0,0,78,0,0,0,0,78],[88,0,0,0,0,88,0,0,0,0,88,1,0,0,0,1],[88,23,0,0,0,1,0,0,0,0,1,88,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[66,86,0,0,87,23,0,0,0,0,88,0,0,0,87,1] >;
C11×C22≀C2 in GAP, Magma, Sage, TeX
C_{11}\times C_2^2\wr C_2 % in TeX
G:=Group("C11xC2^2wrC2"); // GroupNames label
G:=SmallGroup(352,155);
// by ID
G=gap.SmallGroup(352,155);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,3242]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^11=b^2=c^2=d^2=e^2=f^2=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,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations