metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊C4⋊4D10, (C2×C20).10D4, (C22×C4)⋊2D10, (C2×D4).43D10, C23⋊Dic5⋊6C2, (C2×Dic5).4D4, (C22×D5).4D4, C22.34(D4×D5), C10.50C22≀C2, D4⋊6D10.3C2, (C22×C20)⋊2C22, (C22×C10).21D4, C22.D4⋊1D5, C23.9(C5⋊D4), C5⋊2(C23.7D4), C23.D5⋊5C22, (D4×C10).59C22, C23.1D10⋊6C2, C2.18(C23⋊D10), C23.75(C22×D5), C23.23D10⋊1C2, (C22×C10).114C23, (C2×C10).31(C2×D4), (C2×C4).9(C5⋊D4), (C2×C5⋊D4).6C22, C22.30(C2×C5⋊D4), (C5×C22⋊C4)⋊35C22, (C5×C22.D4)⋊1C2, SmallGroup(320,680)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4⋊D10
G = < a,b,c,d,e | a2=b2=c4=d10=e2=1, cac-1=dad-1=ab=ba, ae=ea, bc=cb, bd=db, be=eb, dcd-1=bc-1, ece=abc, ede=d-1 >
Subgroups: 734 in 160 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C23⋊C4, C22.D4, C22.D4, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C23.7D4, C10.D4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, C23.1D10, C23⋊Dic5, C23.23D10, C5×C22.D4, D4⋊6D10, C22⋊C4⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, C23.7D4, D4×D5, C2×C5⋊D4, C23⋊D10, C22⋊C4⋊D10
►On 80 points
Generators in S80
(1 38)(2 34)(3 40)(4 36)(5 32)(6 31)(7 37)(8 33)(9 39)(10 35)(11 67)(12 63)(13 69)(14 65)(15 61)(16 62)(17 68)(18 64)(19 70)(20 66)(21 72)(22 46)(23 74)(24 48)(25 76)(26 50)(27 78)(28 42)(29 80)(30 44)(41 56)(43 58)(45 60)(47 52)(49 54)(51 73)(53 75)(55 77)(57 79)(59 71)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)(21 60)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 36)(32 37)(33 38)(34 39)(35 40)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 51 16 27)(2 57 17 23)(3 53 18 29)(4 59 19 25)(5 55 20 21)(6 30 14 54)(7 26 15 60)(8 22 11 56)(9 28 12 52)(10 24 13 58)(31 71 65 76)(32 50 66 45)(33 73 67 78)(34 42 68 47)(35 75 69 80)(36 44 70 49)(37 77 61 72)(38 46 62 41)(39 79 63 74)(40 48 64 43)
(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 5)(2 4)(6 9)(7 8)(11 20)(12 19)(13 18)(14 17)(15 16)(21 41)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 39)(32 38)(33 37)(34 36)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 80)(59 79)(60 78)(61 62)(63 70)(64 69)(65 68)(66 67)
G:=sub<Sym(80)| (1,38)(2,34)(3,40)(4,36)(5,32)(6,31)(7,37)(8,33)(9,39)(10,35)(11,67)(12,63)(13,69)(14,65)(15,61)(16,62)(17,68)(18,64)(19,70)(20,66)(21,72)(22,46)(23,74)(24,48)(25,76)(26,50)(27,78)(28,42)(29,80)(30,44)(41,56)(43,58)(45,60)(47,52)(49,54)(51,73)(53,75)(55,77)(57,79)(59,71), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,60)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,36)(32,37)(33,38)(34,39)(35,40)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(61,66)(62,67)(63,68)(64,69)(65,70), (1,51,16,27)(2,57,17,23)(3,53,18,29)(4,59,19,25)(5,55,20,21)(6,30,14,54)(7,26,15,60)(8,22,11,56)(9,28,12,52)(10,24,13,58)(31,71,65,76)(32,50,66,45)(33,73,67,78)(34,42,68,47)(35,75,69,80)(36,44,70,49)(37,77,61,72)(38,46,62,41)(39,79,63,74)(40,48,64,43), (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), (1,5)(2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(21,41)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,39)(32,38)(33,37)(34,36)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,80)(59,79)(60,78)(61,62)(63,70)(64,69)(65,68)(66,67)>;
G:=Group( (1,38)(2,34)(3,40)(4,36)(5,32)(6,31)(7,37)(8,33)(9,39)(10,35)(11,67)(12,63)(13,69)(14,65)(15,61)(16,62)(17,68)(18,64)(19,70)(20,66)(21,72)(22,46)(23,74)(24,48)(25,76)(26,50)(27,78)(28,42)(29,80)(30,44)(41,56)(43,58)(45,60)(47,52)(49,54)(51,73)(53,75)(55,77)(57,79)(59,71), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,60)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,36)(32,37)(33,38)(34,39)(35,40)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(61,66)(62,67)(63,68)(64,69)(65,70), (1,51,16,27)(2,57,17,23)(3,53,18,29)(4,59,19,25)(5,55,20,21)(6,30,14,54)(7,26,15,60)(8,22,11,56)(9,28,12,52)(10,24,13,58)(31,71,65,76)(32,50,66,45)(33,73,67,78)(34,42,68,47)(35,75,69,80)(36,44,70,49)(37,77,61,72)(38,46,62,41)(39,79,63,74)(40,48,64,43), (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), (1,5)(2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(21,41)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,39)(32,38)(33,37)(34,36)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,80)(59,79)(60,78)(61,62)(63,70)(64,69)(65,68)(66,67) );
G=PermutationGroup([[(1,38),(2,34),(3,40),(4,36),(5,32),(6,31),(7,37),(8,33),(9,39),(10,35),(11,67),(12,63),(13,69),(14,65),(15,61),(16,62),(17,68),(18,64),(19,70),(20,66),(21,72),(22,46),(23,74),(24,48),(25,76),(26,50),(27,78),(28,42),(29,80),(30,44),(41,56),(43,58),(45,60),(47,52),(49,54),(51,73),(53,75),(55,77),(57,79),(59,71)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,16),(12,17),(13,18),(14,19),(15,20),(21,60),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,36),(32,37),(33,38),(34,39),(35,40),(41,78),(42,79),(43,80),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(61,66),(62,67),(63,68),(64,69),(65,70)], [(1,51,16,27),(2,57,17,23),(3,53,18,29),(4,59,19,25),(5,55,20,21),(6,30,14,54),(7,26,15,60),(8,22,11,56),(9,28,12,52),(10,24,13,58),(31,71,65,76),(32,50,66,45),(33,73,67,78),(34,42,68,47),(35,75,69,80),(36,44,70,49),(37,77,61,72),(38,46,62,41),(39,79,63,74),(40,48,64,43)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,20),(12,19),(13,18),(14,17),(15,16),(21,41),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,39),(32,38),(33,37),(34,36),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,80),(59,79),(60,78),(61,62),(63,70),(64,69),(65,68),(66,67)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 4 | 8 | 20 | 20 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C23.7D4 | D4×D5 | C22⋊C4⋊D10 |
| kernel | C22⋊C4⋊D10 | C23.1D10 | C23⋊Dic5 | C23.23D10 | C5×C22.D4 | D4⋊6D10 | C2×Dic5 | C2×C20 | C22×D5 | C22×C10 | C22.D4 | C22⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C5 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 8 |
Matrix representation of C22⋊C4⋊D10 ►in GL4(𝔽41) generated by
| 36 | 35 | 12 | 12 |
| 21 | 17 | 0 | 14 |
| 6 | 0 | 23 | 6 |
| 23 | 6 | 23 | 6 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 12 | 0 | 12 | 3 |
| 33 | 9 | 9 | 33 |
| 14 | 25 | 32 | 0 |
| 11 | 25 | 29 | 29 |
| 0 | 35 | 0 | 0 |
| 7 | 34 | 0 | 0 |
| 30 | 35 | 6 | 6 |
| 25 | 29 | 35 | 1 |
| 7 | 35 | 0 | 0 |
| 8 | 34 | 0 | 0 |
| 30 | 35 | 6 | 6 |
| 25 | 29 | 1 | 35 |
G:=sub<GL(4,GF(41))| [36,21,6,23,35,17,0,6,12,0,23,23,12,14,6,6],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[12,33,14,11,0,9,25,25,12,9,32,29,3,33,0,29],[0,7,30,25,35,34,35,29,0,0,6,35,0,0,6,1],[7,8,30,25,35,34,35,29,0,0,6,1,0,0,6,35] >;
C22⋊C4⋊D10 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\rtimes D_{10} % in TeX
G:=Group("C2^2:C4:D10"); // GroupNames label
G:=SmallGroup(320,680);
// by ID
G=gap.SmallGroup(320,680);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,570,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^10=e^2=1,c*a*c^-1=d*a*d^-1=a*b=b*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b*c^-1,e*c*e=a*b*c,e*d*e=d^-1>;
// generators/relations