direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C5⋊D4, C24⋊3D5, C23⋊4D10, D10⋊3C23, C10.15C24, Dic5⋊2C23, (C2×C10)⋊9D4, C10⋊3(C2×D4), C5⋊3(C22×D4), (C2×C10)⋊3C23, (C23×C10)⋊4C2, (C23×D5)⋊5C2, C2.15(C23×D5), C22⋊2(C22×D5), (C22×C10)⋊7C22, (C22×Dic5)⋊9C2, (C22×D5)⋊7C22, (C2×Dic5)⋊11C22, SmallGroup(160,227)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C5⋊D4
G = < a,b,c,d,e | a2=b2=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 680 in 236 conjugacy classes, 105 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22×C4, C2×D4, C24, C24, Dic5, D10, D10, C2×C10, C2×C10, C22×D4, C2×Dic5, C5⋊D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C22×Dic5, C2×C5⋊D4, C23×D5, C23×C10, C22×C5⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C5⋊D4, C22×D5, C2×C5⋊D4, C23×D5, C22×C5⋊D4
►On 80 points
(1 46)(2 47)(3 48)(4 49)(5 50)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
(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)
(1 51 6 56)(2 55 7 60)(3 54 8 59)(4 53 9 58)(5 52 10 57)(11 46 16 41)(12 50 17 45)(13 49 18 44)(14 48 19 43)(15 47 20 42)(21 71 26 76)(22 75 27 80)(23 74 28 79)(24 73 29 78)(25 72 30 77)(31 66 36 61)(32 70 37 65)(33 69 38 64)(34 68 39 63)(35 67 40 62)
(1 61)(2 65)(3 64)(4 63)(5 62)(6 66)(7 70)(8 69)(9 68)(10 67)(11 76)(12 80)(13 79)(14 78)(15 77)(16 71)(17 75)(18 74)(19 73)(20 72)(21 41)(22 45)(23 44)(24 43)(25 42)(26 46)(27 50)(28 49)(29 48)(30 47)(31 56)(32 60)(33 59)(34 58)(35 57)(36 51)(37 55)(38 54)(39 53)(40 52)
G:=sub<Sym(80)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75), (1,26)(2,27)(3,28)(4,29)(5,30)(6,21)(7,22)(8,23)(9,24)(10,25)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75), (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), (1,51,6,56)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,46,16,41)(12,50,17,45)(13,49,18,44)(14,48,19,43)(15,47,20,42)(21,71,26,76)(22,75,27,80)(23,74,28,79)(24,73,29,78)(25,72,30,77)(31,66,36,61)(32,70,37,65)(33,69,38,64)(34,68,39,63)(35,67,40,62), (1,61)(2,65)(3,64)(4,63)(5,62)(6,66)(7,70)(8,69)(9,68)(10,67)(11,76)(12,80)(13,79)(14,78)(15,77)(16,71)(17,75)(18,74)(19,73)(20,72)(21,41)(22,45)(23,44)(24,43)(25,42)(26,46)(27,50)(28,49)(29,48)(30,47)(31,56)(32,60)(33,59)(34,58)(35,57)(36,51)(37,55)(38,54)(39,53)(40,52)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75), (1,26)(2,27)(3,28)(4,29)(5,30)(6,21)(7,22)(8,23)(9,24)(10,25)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75), (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), (1,51,6,56)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,46,16,41)(12,50,17,45)(13,49,18,44)(14,48,19,43)(15,47,20,42)(21,71,26,76)(22,75,27,80)(23,74,28,79)(24,73,29,78)(25,72,30,77)(31,66,36,61)(32,70,37,65)(33,69,38,64)(34,68,39,63)(35,67,40,62), (1,61)(2,65)(3,64)(4,63)(5,62)(6,66)(7,70)(8,69)(9,68)(10,67)(11,76)(12,80)(13,79)(14,78)(15,77)(16,71)(17,75)(18,74)(19,73)(20,72)(21,41)(22,45)(23,44)(24,43)(25,42)(26,46)(27,50)(28,49)(29,48)(30,47)(31,56)(32,60)(33,59)(34,58)(35,57)(36,51)(37,55)(38,54)(39,53)(40,52) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,41),(7,42),(8,43),(9,44),(10,45),(11,56),(12,57),(13,58),(14,59),(15,60),(16,51),(17,52),(18,53),(19,54),(20,55),(21,66),(22,67),(23,68),(24,69),(25,70),(26,61),(27,62),(28,63),(29,64),(30,65),(31,76),(32,77),(33,78),(34,79),(35,80),(36,71),(37,72),(38,73),(39,74),(40,75)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,21),(7,22),(8,23),(9,24),(10,25),(11,36),(12,37),(13,38),(14,39),(15,40),(16,31),(17,32),(18,33),(19,34),(20,35),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)], [(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)], [(1,51,6,56),(2,55,7,60),(3,54,8,59),(4,53,9,58),(5,52,10,57),(11,46,16,41),(12,50,17,45),(13,49,18,44),(14,48,19,43),(15,47,20,42),(21,71,26,76),(22,75,27,80),(23,74,28,79),(24,73,29,78),(25,72,30,77),(31,66,36,61),(32,70,37,65),(33,69,38,64),(34,68,39,63),(35,67,40,62)], [(1,61),(2,65),(3,64),(4,63),(5,62),(6,66),(7,70),(8,69),(9,68),(10,67),(11,76),(12,80),(13,79),(14,78),(15,77),(16,71),(17,75),(18,74),(19,73),(20,72),(21,41),(22,45),(23,44),(24,43),(25,42),(26,46),(27,50),(28,49),(29,48),(30,47),(31,56),(32,60),(33,59),(34,58),(35,57),(36,51),(37,55),(38,54),(39,53),(40,52)]])
C22×C5⋊D4 is a maximal subgroup of
C24.12D10 C24.13D10 C23.45D20 C24.14D10 C23⋊2D20 C24.16D10 C24.65D10 C24.21D10 C24.24D10 C24.27D10 C23⋊3D20 C24⋊3D10 C24⋊4D10 C24.33D10 C24.34D10 C24⋊8D10 C22×D4×D5
C22×C5⋊D4 is a maximal quotient of
C24.72D10 C24⋊8D10 C24.41D10 C24.42D10 C10.442- 1+4 C10.452- 1+4 C20.C24 C10.1042- 1+4 C10.1052- 1+4 (C2×C20)⋊15D4 C10.1452+ 1+4 C10.1462+ 1+4 C10.1072- 1+4 (C2×C20)⋊17D4 C10.1472+ 1+4 C10.1482+ 1+4 D20.32C23 D20.33C23 D20.34C23 D20.35C23
52 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 5A | 5B | 10A | ··· | 10AD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | C5⋊D4 |
| kernel | C22×C5⋊D4 | C22×Dic5 | C2×C5⋊D4 | C23×D5 | C23×C10 | C2×C10 | C24 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 4 | 2 | 14 | 16 |
Matrix representation of C22×C5⋊D4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 40 |
| 0 | 0 | 8 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 3 | 18 |
| 0 | 0 | 4 | 38 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 1 |
| 0 | 0 | 34 | 7 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,7,8,0,0,40,40],[1,0,0,0,0,40,0,0,0,0,3,4,0,0,18,38],[40,0,0,0,0,1,0,0,0,0,34,34,0,0,1,7] >;
C22×C5⋊D4 in GAP, Magma, Sage, TeX
C_2^2\times C_5\rtimes D_4
% in TeX
G:=Group("C2^2xC5:D4"); // GroupNames label
G:=SmallGroup(160,227);
// by ID
G=gap.SmallGroup(160,227);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,579,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations