metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D22⋊5D4, C23⋊1D22, (C2×C22)⋊2D4, (C2×C4)⋊2D22, C11⋊2C22≀C2, (D4×C22)⋊8C2, (C2×D4)⋊3D11, D22⋊C4⋊14C2, (C2×C44)⋊7C22, C22.49(C2×D4), C2.25(D4×D11), (C23×D11)⋊2C2, C22⋊2(C11⋊D4), (C2×C22).52C23, (C22×C22)⋊3C22, C23.D11⋊10C2, (C2×Dic11)⋊2C22, C22.59(C22×D11), (C22×D11).25C22, (C2×C11⋊D4)⋊4C2, C2.13(C2×C11⋊D4), SmallGroup(352,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊D22
G = < a,b,c,d,e | a2=b2=c2=d22=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 922 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C2×D4, C2×D4, C24, D11, C22, C22, C22, C22≀C2, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×C22, C2×Dic11, C11⋊D4, C2×C44, D4×C11, C22×D11, C22×D11, C22×C22, D22⋊C4, C23.D11, C2×C11⋊D4, D4×C22, C23×D11, C23⋊D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C22≀C2, D22, C11⋊D4, C22×D11, D4×D11, C2×C11⋊D4, C23⋊D22
►On 88 points
Generators in S88
(1 87)(2 24)(3 67)(4 26)(5 69)(6 28)(7 71)(8 30)(9 73)(10 32)(11 75)(12 34)(13 77)(14 36)(15 79)(16 38)(17 81)(18 40)(19 83)(20 42)(21 85)(22 44)(23 63)(25 65)(27 45)(29 47)(31 49)(33 51)(35 53)(37 55)(39 57)(41 59)(43 61)(46 70)(48 72)(50 74)(52 76)(54 78)(56 80)(58 82)(60 84)(62 86)(64 88)(66 68)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)
(1 63)(2 64)(3 65)(4 66)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 87)(24 88)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)
(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 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 66)(20 65)(21 64)(22 63)(23 75)(24 74)(25 73)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)
G:=sub<Sym(88)| (1,87)(2,24)(3,67)(4,26)(5,69)(6,28)(7,71)(8,30)(9,73)(10,32)(11,75)(12,34)(13,77)(14,36)(15,79)(16,38)(17,81)(18,40)(19,83)(20,42)(21,85)(22,44)(23,63)(25,65)(27,45)(29,47)(31,49)(33,51)(35,53)(37,55)(39,57)(41,59)(43,61)(46,70)(48,72)(50,74)(52,76)(54,78)(56,80)(58,82)(60,84)(62,86)(64,88)(66,68), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (1,63)(2,64)(3,65)(4,66)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,87)(24,88)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86), (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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,66)(20,65)(21,64)(22,63)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)>;
G:=Group( (1,87)(2,24)(3,67)(4,26)(5,69)(6,28)(7,71)(8,30)(9,73)(10,32)(11,75)(12,34)(13,77)(14,36)(15,79)(16,38)(17,81)(18,40)(19,83)(20,42)(21,85)(22,44)(23,63)(25,65)(27,45)(29,47)(31,49)(33,51)(35,53)(37,55)(39,57)(41,59)(43,61)(46,70)(48,72)(50,74)(52,76)(54,78)(56,80)(58,82)(60,84)(62,86)(64,88)(66,68), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (1,63)(2,64)(3,65)(4,66)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,87)(24,88)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86), (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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,66)(20,65)(21,64)(22,63)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76) );
G=PermutationGroup([[(1,87),(2,24),(3,67),(4,26),(5,69),(6,28),(7,71),(8,30),(9,73),(10,32),(11,75),(12,34),(13,77),(14,36),(15,79),(16,38),(17,81),(18,40),(19,83),(20,42),(21,85),(22,44),(23,63),(25,65),(27,45),(29,47),(31,49),(33,51),(35,53),(37,55),(39,57),(41,59),(43,61),(46,70),(48,72),(50,74),(52,76),(54,78),(56,80),(58,82),(60,84),(62,86),(64,88),(66,68)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88)], [(1,63),(2,64),(3,65),(4,66),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,87),(24,88),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,85),(44,86)], [(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)], [(1,62),(2,61),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,66),(20,65),(21,64),(22,63),(23,75),(24,74),(25,73),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22AI | 44A | ··· | 44J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 22 | 22 | 22 | 22 | 4 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D11 | D22 | D22 | C11⋊D4 | D4×D11 |
| kernel | C23⋊D22 | D22⋊C4 | C23.D11 | C2×C11⋊D4 | D4×C22 | C23×D11 | D22 | C2×C22 | C2×D4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 5 | 5 | 10 | 20 | 10 |
Matrix representation of C23⋊D22 ►in GL4(𝔽89) generated by
| 3 | 39 | 0 | 0 |
| 50 | 86 | 0 | 0 |
| 0 | 0 | 68 | 42 |
| 0 | 0 | 87 | 21 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 62 | 30 | 0 | 0 |
| 59 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 88 |
| 30 | 62 | 0 | 0 |
| 30 | 59 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
G:=sub<GL(4,GF(89))| [3,50,0,0,39,86,0,0,0,0,68,87,0,0,42,21],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[62,59,0,0,30,30,0,0,0,0,1,1,0,0,0,88],[30,30,0,0,62,59,0,0,0,0,88,0,0,0,0,88] >;
C23⋊D22 in GAP, Magma, Sage, TeX
C_2^3\rtimes D_{22} % in TeX
G:=Group("C2^3:D22"); // GroupNames label
G:=SmallGroup(352,132);
// by ID
G=gap.SmallGroup(352,132);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,188,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^22=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations