metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊36D10, (C2×D20)⋊23C4, (C2×C4).46D20, C4.62(C2×D20), D20.39(C2×C4), C20.142(C2×D4), (C2×C20).472D4, C42⋊C2⋊1D5, D20⋊6C4⋊27C2, (C22×C10).74D4, C2.2(D4⋊D10), C20.94(C22⋊C4), C20.123(C22×C4), (C2×C20).328C23, (C22×D20).13C2, (C22×C4).110D10, C23.54(C5⋊D4), C5⋊3(C23.37D4), C4.23(D10⋊C4), C10.105(C8⋊C22), (C2×D20).242C22, (C22×C20).150C22, C22.23(D10⋊C4), C4.51(C2×C4×D5), (C2×C4).44(C4×D5), (C5×C4⋊C4)⋊41C22, (C2×C5⋊2C8)⋊4C22, (C2×C4.Dic5)⋊9C2, (C2×C20).264(C2×C4), (C5×C42⋊C2)⋊1C2, (C2×C10).457(C2×D4), C10.85(C2×C22⋊C4), C22.72(C2×C5⋊D4), C2.17(C2×D10⋊C4), (C2×C4).241(C5⋊D4), (C2×C4).428(C22×D5), (C2×C10).78(C22⋊C4), SmallGroup(320,628)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊36D10
G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab-1, dcd=c-1 >
Subgroups: 974 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C24, C20, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C5⋊2C8, D20, D20, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C23.37D4, C2×C5⋊2C8, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×D20, C22×C20, C23×D5, D20⋊6C4, C2×C4.Dic5, C5×C42⋊C2, C22×D20, C4⋊C4⋊36D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8⋊C22, C4×D5, D20, C5⋊D4, C22×D5, C23.37D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4⋊D10, C4⋊C4⋊36D10
►On 80 points
(1 20 33 50)(2 11 34 41)(3 12 35 42)(4 13 36 43)(5 14 37 44)(6 15 38 45)(7 16 39 46)(8 17 40 47)(9 18 31 48)(10 19 32 49)(21 68 55 73)(22 69 56 74)(23 70 57 75)(24 61 58 76)(25 62 59 77)(26 63 60 78)(27 64 51 79)(28 65 52 80)(29 66 53 71)(30 67 54 72)
(1 73 38 78)(2 69 39 64)(3 75 40 80)(4 61 31 66)(5 77 32 72)(6 63 33 68)(7 79 34 74)(8 65 35 70)(9 71 36 76)(10 67 37 62)(11 22 46 27)(12 57 47 52)(13 24 48 29)(14 59 49 54)(15 26 50 21)(16 51 41 56)(17 28 42 23)(18 53 43 58)(19 30 44 25)(20 55 45 60)
(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 10)(7 9)(11 43)(12 42)(13 41)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 70)(29 69)(30 68)(31 39)(32 38)(33 37)(34 36)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 80)(58 79)(59 78)(60 77)
G:=sub<Sym(80)| (1,20,33,50)(2,11,34,41)(3,12,35,42)(4,13,36,43)(5,14,37,44)(6,15,38,45)(7,16,39,46)(8,17,40,47)(9,18,31,48)(10,19,32,49)(21,68,55,73)(22,69,56,74)(23,70,57,75)(24,61,58,76)(25,62,59,77)(26,63,60,78)(27,64,51,79)(28,65,52,80)(29,66,53,71)(30,67,54,72), (1,73,38,78)(2,69,39,64)(3,75,40,80)(4,61,31,66)(5,77,32,72)(6,63,33,68)(7,79,34,74)(8,65,35,70)(9,71,36,76)(10,67,37,62)(11,22,46,27)(12,57,47,52)(13,24,48,29)(14,59,49,54)(15,26,50,21)(16,51,41,56)(17,28,42,23)(18,53,43,58)(19,30,44,25)(20,55,45,60), (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)(6,10)(7,9)(11,43)(12,42)(13,41)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,70)(29,69)(30,68)(31,39)(32,38)(33,37)(34,36)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77)>;
G:=Group( (1,20,33,50)(2,11,34,41)(3,12,35,42)(4,13,36,43)(5,14,37,44)(6,15,38,45)(7,16,39,46)(8,17,40,47)(9,18,31,48)(10,19,32,49)(21,68,55,73)(22,69,56,74)(23,70,57,75)(24,61,58,76)(25,62,59,77)(26,63,60,78)(27,64,51,79)(28,65,52,80)(29,66,53,71)(30,67,54,72), (1,73,38,78)(2,69,39,64)(3,75,40,80)(4,61,31,66)(5,77,32,72)(6,63,33,68)(7,79,34,74)(8,65,35,70)(9,71,36,76)(10,67,37,62)(11,22,46,27)(12,57,47,52)(13,24,48,29)(14,59,49,54)(15,26,50,21)(16,51,41,56)(17,28,42,23)(18,53,43,58)(19,30,44,25)(20,55,45,60), (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)(6,10)(7,9)(11,43)(12,42)(13,41)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,70)(29,69)(30,68)(31,39)(32,38)(33,37)(34,36)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77) );
G=PermutationGroup([[(1,20,33,50),(2,11,34,41),(3,12,35,42),(4,13,36,43),(5,14,37,44),(6,15,38,45),(7,16,39,46),(8,17,40,47),(9,18,31,48),(10,19,32,49),(21,68,55,73),(22,69,56,74),(23,70,57,75),(24,61,58,76),(25,62,59,77),(26,63,60,78),(27,64,51,79),(28,65,52,80),(29,66,53,71),(30,67,54,72)], [(1,73,38,78),(2,69,39,64),(3,75,40,80),(4,61,31,66),(5,77,32,72),(6,63,33,68),(7,79,34,74),(8,65,35,70),(9,71,36,76),(10,67,37,62),(11,22,46,27),(12,57,47,52),(13,24,48,29),(14,59,49,54),(15,26,50,21),(16,51,41,56),(17,28,42,23),(18,53,43,58),(19,30,44,25),(20,55,45,60)], [(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),(6,10),(7,9),(11,43),(12,42),(13,41),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,70),(29,69),(30,68),(31,39),(32,38),(33,37),(34,36),(51,76),(52,75),(53,74),(54,73),(55,72),(56,71),(57,80),(58,79),(59,78),(60,77)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4⋊D10 |
| kernel | C4⋊C4⋊36D10 | D20⋊6C4 | C2×C4.Dic5 | C5×C42⋊C2 | C22×D20 | C2×D20 | C2×C20 | C22×C10 | C42⋊C2 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 2 | 8 |
Matrix representation of C4⋊C4⋊36D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 40 |
| 0 | 0 | 37 | 0 | 1 | 0 |
| 11 | 32 | 0 | 0 | 0 | 0 |
| 9 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 34 | 37 | 0 |
| 0 | 0 | 34 | 8 | 0 | 37 |
| 0 | 0 | 28 | 13 | 33 | 7 |
| 0 | 0 | 13 | 28 | 7 | 33 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 37 | 24 | 1 | 0 |
| 0 | 0 | 24 | 37 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 31 | 14 | 0 | 40 |
| 0 | 0 | 31 | 27 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[11,9,0,0,0,0,32,30,0,0,0,0,0,0,8,34,28,13,0,0,34,8,13,28,0,0,37,0,33,7,0,0,0,37,7,33],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,37,24,0,0,0,40,24,37,0,0,0,0,1,0,0,0,0,0,0,1],[1,7,0,0,0,0,0,40,0,0,0,0,0,0,1,0,31,31,0,0,0,40,14,27,0,0,0,0,0,40,0,0,0,0,40,0] >;
C4⋊C4⋊36D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{36}D_{10} % in TeX
G:=Group("C4:C4:36D10"); // GroupNames label
G:=SmallGroup(320,628);
// by ID
G=gap.SmallGroup(320,628);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations