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 [×6], C4 [×7], C22, C22 [×2], C22 [×8], C5, C2×C4, C2×C4 [×11], D4 [×9], Q8, C23 [×2], C23 [×3], D5 [×2], C10, C10 [×4], C22⋊C4, C22⋊C4 [×5], C4⋊C4 [×3], C22×C4, C2×D4, C2×D4 [×5], C4○D4 [×3], Dic5 [×4], C20 [×3], D10 [×5], C2×C10, C2×C10 [×2], C2×C10 [×3], C23⋊C4 [×3], C22.D4, C22.D4 [×2], 2+ 1+4, Dic10, C4×D5 [×2], D20, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20, C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×D5, C22×C10 [×2], C23.7D4, C10.D4 [×2], D10⋊C4 [×2], C23.D5 [×2], C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C4○D20, D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×2], C2×C5⋊D4, C22×C20, D4×C10, C23.1D10 [×2], C23⋊Dic5, C23.23D10 [×2], C5×C22.D4, D4⋊6D10, C22⋊C4⋊D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, C23.7D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C22⋊C4⋊D10
►On 80 points
(1 39)(2 35)(3 31)(4 37)(5 33)(6 36)(7 32)(8 38)(9 34)(10 40)(11 26)(12 22)(13 28)(14 24)(15 30)(16 29)(17 25)(18 21)(19 27)(20 23)(41 78)(42 66)(43 80)(44 68)(45 72)(46 70)(47 74)(48 62)(49 76)(50 64)(51 63)(52 77)(53 65)(54 79)(55 67)(56 71)(57 69)(58 73)(59 61)(60 75)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 18)(12 19)(13 20)(14 16)(15 17)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 51)(50 52)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)
(1 53 18 46)(2 59 19 42)(3 55 20 48)(4 51 16 44)(5 57 17 50)(6 43 13 60)(7 49 14 56)(8 45 15 52)(9 41 11 58)(10 47 12 54)(21 73 39 78)(22 66 40 61)(23 75 31 80)(24 68 32 63)(25 77 33 72)(26 70 34 65)(27 79 35 74)(28 62 36 67)(29 71 37 76)(30 64 38 69)
(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)(7 10)(8 9)(11 17)(12 16)(13 20)(14 19)(15 18)(21 30)(22 29)(23 28)(24 27)(25 26)(32 40)(33 39)(34 38)(35 37)(41 72)(42 71)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)
G:=sub<Sym(80)| (1,39)(2,35)(3,31)(4,37)(5,33)(6,36)(7,32)(8,38)(9,34)(10,40)(11,26)(12,22)(13,28)(14,24)(15,30)(16,29)(17,25)(18,21)(19,27)(20,23)(41,78)(42,66)(43,80)(44,68)(45,72)(46,70)(47,74)(48,62)(49,76)(50,64)(51,63)(52,77)(53,65)(54,79)(55,67)(56,71)(57,69)(58,73)(59,61)(60,75), (1,9)(2,10)(3,6)(4,7)(5,8)(11,18)(12,19)(13,20)(14,16)(15,17)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,51)(50,52)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73), (1,53,18,46)(2,59,19,42)(3,55,20,48)(4,51,16,44)(5,57,17,50)(6,43,13,60)(7,49,14,56)(8,45,15,52)(9,41,11,58)(10,47,12,54)(21,73,39,78)(22,66,40,61)(23,75,31,80)(24,68,32,63)(25,77,33,72)(26,70,34,65)(27,79,35,74)(28,62,36,67)(29,71,37,76)(30,64,38,69), (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)(7,10)(8,9)(11,17)(12,16)(13,20)(14,19)(15,18)(21,30)(22,29)(23,28)(24,27)(25,26)(32,40)(33,39)(34,38)(35,37)(41,72)(42,71)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)>;
G:=Group( (1,39)(2,35)(3,31)(4,37)(5,33)(6,36)(7,32)(8,38)(9,34)(10,40)(11,26)(12,22)(13,28)(14,24)(15,30)(16,29)(17,25)(18,21)(19,27)(20,23)(41,78)(42,66)(43,80)(44,68)(45,72)(46,70)(47,74)(48,62)(49,76)(50,64)(51,63)(52,77)(53,65)(54,79)(55,67)(56,71)(57,69)(58,73)(59,61)(60,75), (1,9)(2,10)(3,6)(4,7)(5,8)(11,18)(12,19)(13,20)(14,16)(15,17)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,51)(50,52)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73), (1,53,18,46)(2,59,19,42)(3,55,20,48)(4,51,16,44)(5,57,17,50)(6,43,13,60)(7,49,14,56)(8,45,15,52)(9,41,11,58)(10,47,12,54)(21,73,39,78)(22,66,40,61)(23,75,31,80)(24,68,32,63)(25,77,33,72)(26,70,34,65)(27,79,35,74)(28,62,36,67)(29,71,37,76)(30,64,38,69), (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)(7,10)(8,9)(11,17)(12,16)(13,20)(14,19)(15,18)(21,30)(22,29)(23,28)(24,27)(25,26)(32,40)(33,39)(34,38)(35,37)(41,72)(42,71)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62) );
G=PermutationGroup([(1,39),(2,35),(3,31),(4,37),(5,33),(6,36),(7,32),(8,38),(9,34),(10,40),(11,26),(12,22),(13,28),(14,24),(15,30),(16,29),(17,25),(18,21),(19,27),(20,23),(41,78),(42,66),(43,80),(44,68),(45,72),(46,70),(47,74),(48,62),(49,76),(50,64),(51,63),(52,77),(53,65),(54,79),(55,67),(56,71),(57,69),(58,73),(59,61),(60,75)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,18),(12,19),(13,20),(14,16),(15,17),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60),(49,51),(50,52),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73)], [(1,53,18,46),(2,59,19,42),(3,55,20,48),(4,51,16,44),(5,57,17,50),(6,43,13,60),(7,49,14,56),(8,45,15,52),(9,41,11,58),(10,47,12,54),(21,73,39,78),(22,66,40,61),(23,75,31,80),(24,68,32,63),(25,77,33,72),(26,70,34,65),(27,79,35,74),(28,62,36,67),(29,71,37,76),(30,64,38,69)], [(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),(7,10),(8,9),(11,17),(12,16),(13,20),(14,19),(15,18),(21,30),(22,29),(23,28),(24,27),(25,26),(32,40),(33,39),(34,38),(35,37),(41,72),(42,71),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,61),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62)])
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