metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊24D10, C10.1382+ 1+4, C4⋊C4⋊34D10, (C4×D20)⋊14C2, (C4×C20)⋊8C22, C42⋊D5⋊6C2, C42⋊2C2⋊3D5, D20⋊8C4⋊40C2, D10⋊2Q8⋊40C2, D10⋊Q8⋊41C2, C22⋊D20.4C2, C4⋊Dic5⋊62C22, C22⋊C4.77D10, D10.40(C4○D4), (C2×C20).603C23, (C2×C10).249C24, (C4×Dic5)⋊58C22, D10.13D4⋊39C2, D10.12D4⋊49C2, C2.63(D4⋊8D10), D10⋊C4⋊63C22, C23.55(C22×D5), Dic5.5D4⋊45C2, C5⋊9(C22.45C24), (C2×Dic10)⋊33C22, (C2×D20).234C22, C23.D10⋊45C2, C10.D4⋊68C22, (C22×C10).63C23, (C23×D5).69C22, C22.270(C23×D5), C23.D5.65C22, (C2×Dic5).129C23, (C22×D5).233C23, C2.96(D5×C4○D4), (C2×C4×D5)⋊53C22, C4⋊C4⋊7D5⋊39C2, (C5×C4⋊C4)⋊33C22, (D5×C22⋊C4)⋊21C2, (C5×C42⋊2C2)⋊4C2, C10.207(C2×C4○D4), (C2×C4).86(C22×D5), (C2×C5⋊D4).69C22, (C5×C22⋊C4).74C22, SmallGroup(320,1377)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊24D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
Subgroups: 1014 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, C22.45C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C2×C5⋊D4, C23×D5, C42⋊D5, C4×D20, C23.D10, D5×C22⋊C4, C22⋊D20, D10.12D4, Dic5.5D4, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4, D10⋊Q8, D10⋊2Q8, C5×C42⋊2C2, C42⋊24D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.45C24, C23×D5, D5×C4○D4, D4⋊8D10, C42⋊24D10
►On 80 points
(1 53 9 43)(2 59 10 49)(3 55 6 45)(4 51 7 41)(5 57 8 47)(11 56 16 46)(12 52 17 42)(13 58 18 48)(14 54 19 44)(15 60 20 50)(21 70 26 73)(22 79 27 66)(23 62 28 75)(24 71 29 68)(25 64 30 77)(31 78 36 65)(32 61 37 74)(33 80 38 67)(34 63 39 76)(35 72 40 69)
(1 36 18 21)(2 32 19 27)(3 38 20 23)(4 34 16 29)(5 40 17 25)(6 33 15 28)(7 39 11 24)(8 35 12 30)(9 31 13 26)(10 37 14 22)(41 76 56 71)(42 64 57 69)(43 78 58 73)(44 66 59 61)(45 80 60 75)(46 68 51 63)(47 72 52 77)(48 70 53 65)(49 74 54 79)(50 62 55 67)
(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 14)(12 13)(16 19)(17 18)(21 35)(22 34)(23 33)(24 32)(25 31)(26 40)(27 39)(28 38)(29 37)(30 36)(41 44)(42 43)(45 50)(46 49)(47 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 63)(64 70)(65 69)(66 68)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,53,9,43)(2,59,10,49)(3,55,6,45)(4,51,7,41)(5,57,8,47)(11,56,16,46)(12,52,17,42)(13,58,18,48)(14,54,19,44)(15,60,20,50)(21,70,26,73)(22,79,27,66)(23,62,28,75)(24,71,29,68)(25,64,30,77)(31,78,36,65)(32,61,37,74)(33,80,38,67)(34,63,39,76)(35,72,40,69), (1,36,18,21)(2,32,19,27)(3,38,20,23)(4,34,16,29)(5,40,17,25)(6,33,15,28)(7,39,11,24)(8,35,12,30)(9,31,13,26)(10,37,14,22)(41,76,56,71)(42,64,57,69)(43,78,58,73)(44,66,59,61)(45,80,60,75)(46,68,51,63)(47,72,52,77)(48,70,53,65)(49,74,54,79)(50,62,55,67), (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)(11,14)(12,13)(16,19)(17,18)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,63)(64,70)(65,69)(66,68)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,53,9,43)(2,59,10,49)(3,55,6,45)(4,51,7,41)(5,57,8,47)(11,56,16,46)(12,52,17,42)(13,58,18,48)(14,54,19,44)(15,60,20,50)(21,70,26,73)(22,79,27,66)(23,62,28,75)(24,71,29,68)(25,64,30,77)(31,78,36,65)(32,61,37,74)(33,80,38,67)(34,63,39,76)(35,72,40,69), (1,36,18,21)(2,32,19,27)(3,38,20,23)(4,34,16,29)(5,40,17,25)(6,33,15,28)(7,39,11,24)(8,35,12,30)(9,31,13,26)(10,37,14,22)(41,76,56,71)(42,64,57,69)(43,78,58,73)(44,66,59,61)(45,80,60,75)(46,68,51,63)(47,72,52,77)(48,70,53,65)(49,74,54,79)(50,62,55,67), (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)(11,14)(12,13)(16,19)(17,18)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,63)(64,70)(65,69)(66,68)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,53,9,43),(2,59,10,49),(3,55,6,45),(4,51,7,41),(5,57,8,47),(11,56,16,46),(12,52,17,42),(13,58,18,48),(14,54,19,44),(15,60,20,50),(21,70,26,73),(22,79,27,66),(23,62,28,75),(24,71,29,68),(25,64,30,77),(31,78,36,65),(32,61,37,74),(33,80,38,67),(34,63,39,76),(35,72,40,69)], [(1,36,18,21),(2,32,19,27),(3,38,20,23),(4,34,16,29),(5,40,17,25),(6,33,15,28),(7,39,11,24),(8,35,12,30),(9,31,13,26),(10,37,14,22),(41,76,56,71),(42,64,57,69),(43,78,58,73),(44,66,59,61),(45,80,60,75),(46,68,51,63),(47,72,52,77),(48,70,53,65),(49,74,54,79),(50,62,55,67)], [(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),(11,14),(12,13),(16,19),(17,18),(21,35),(22,34),(23,33),(24,32),(25,31),(26,40),(27,39),(28,38),(29,37),(30,36),(41,44),(42,43),(45,50),(46,49),(47,48),(51,54),(52,53),(55,60),(56,59),(57,58),(61,63),(64,70),(65,69),(66,68),(71,79),(72,78),(73,77),(74,76)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ 1+4 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42⋊24D10 | C42⋊D5 | C4×D20 | C23.D10 | D5×C22⋊C4 | C22⋊D20 | D10.12D4 | Dic5.5D4 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | D10⋊Q8 | D10⋊2Q8 | C5×C42⋊2C2 | C42⋊2C2 | D10 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 2 | 6 | 6 | 1 | 8 | 4 |
Matrix representation of C42⋊24D10 ►in GL6(𝔽41)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C42⋊24D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{24}D_{10} % in TeX
G:=Group("C4^2:24D10"); // GroupNames label
G:=SmallGroup(320,1377);
// by ID
G=gap.SmallGroup(320,1377);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,570,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations