metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊23D10, C10.1372+ 1+4, C4⋊C4⋊33D10, (C4×D20)⋊13C2, (C4×C20)⋊7C22, C4⋊D20⋊36C2, C4.D20⋊8C2, C42⋊2C2⋊2D5, D20⋊8C4⋊39C2, C22⋊D20⋊27C2, D10⋊D4⋊44C2, D10⋊Q8⋊40C2, (C2×D20)⋊29C22, C4⋊Dic5⋊61C22, C22⋊C4.40D10, D10.18(C4○D4), D10⋊C4⋊7C22, (C2×C20).193C23, (C2×C10).248C24, C5⋊9(C22.32C24), (C4×Dic5)⋊38C22, D10.13D4⋊38C2, D10.12D4⋊48C2, C2.62(D4⋊8D10), C23.54(C22×D5), Dic5.5D4⋊44C2, (C2×Dic10)⋊11C22, C10.D4⋊27C22, (C22×C10).62C23, (C23×D5).68C22, C22.269(C23×D5), C23.D5.64C22, (C2×Dic5).274C23, (C22×D5).111C23, C2.95(D5×C4○D4), (C2×C4×D5)⋊27C22, C4⋊C4⋊D5⋊41C2, (C5×C4⋊C4)⋊32C22, (D5×C22⋊C4)⋊20C2, (C5×C42⋊2C2)⋊3C2, C10.206(C2×C4○D4), (C2×C4).85(C22×D5), (C2×C5⋊D4).68C22, (C5×C22⋊C4).73C22, SmallGroup(320,1376)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊23D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >
Subgroups: 1134 in 250 conjugacy classes, 93 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, C4×D4, C22≀C2, C4⋊D4, 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, C22.32C24, 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, C4×D20, C4.D20, D5×C22⋊C4, C22⋊D20, D10.12D4, D10⋊D4, Dic5.5D4, D20⋊8C4, D10.13D4, C4⋊D20, D10⋊Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42⋊23D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, C23×D5, D5×C4○D4, D4⋊8D10, C42⋊23D10
►On 80 points
(1 53 6 43)(2 59 7 49)(3 55 8 45)(4 51 9 41)(5 57 10 47)(11 56 17 46)(12 52 18 42)(13 58 19 48)(14 54 20 44)(15 60 16 50)(21 64 26 77)(22 73 27 70)(23 66 28 79)(24 75 29 62)(25 68 30 71)(31 76 36 63)(32 69 37 72)(33 78 38 65)(34 61 39 74)(35 80 40 67)
(1 38 19 27)(2 34 20 23)(3 40 16 29)(4 36 17 25)(5 32 18 21)(6 33 13 22)(7 39 14 28)(8 35 15 24)(9 31 11 30)(10 37 12 26)(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 33)(22 32)(23 31)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(41 54)(42 53)(43 52)(44 51)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(61 63)(64 70)(65 69)(66 68)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,53,6,43)(2,59,7,49)(3,55,8,45)(4,51,9,41)(5,57,10,47)(11,56,17,46)(12,52,18,42)(13,58,19,48)(14,54,20,44)(15,60,16,50)(21,64,26,77)(22,73,27,70)(23,66,28,79)(24,75,29,62)(25,68,30,71)(31,76,36,63)(32,69,37,72)(33,78,38,65)(34,61,39,74)(35,80,40,67), (1,38,19,27)(2,34,20,23)(3,40,16,29)(4,36,17,25)(5,32,18,21)(6,33,13,22)(7,39,14,28)(8,35,15,24)(9,31,11,30)(10,37,12,26)(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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,33)(22,32)(23,31)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(41,54)(42,53)(43,52)(44,51)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(61,63)(64,70)(65,69)(66,68)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,53,6,43)(2,59,7,49)(3,55,8,45)(4,51,9,41)(5,57,10,47)(11,56,17,46)(12,52,18,42)(13,58,19,48)(14,54,20,44)(15,60,16,50)(21,64,26,77)(22,73,27,70)(23,66,28,79)(24,75,29,62)(25,68,30,71)(31,76,36,63)(32,69,37,72)(33,78,38,65)(34,61,39,74)(35,80,40,67), (1,38,19,27)(2,34,20,23)(3,40,16,29)(4,36,17,25)(5,32,18,21)(6,33,13,22)(7,39,14,28)(8,35,15,24)(9,31,11,30)(10,37,12,26)(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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,33)(22,32)(23,31)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(41,54)(42,53)(43,52)(44,51)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(61,63)(64,70)(65,69)(66,68)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,53,6,43),(2,59,7,49),(3,55,8,45),(4,51,9,41),(5,57,10,47),(11,56,17,46),(12,52,18,42),(13,58,19,48),(14,54,20,44),(15,60,16,50),(21,64,26,77),(22,73,27,70),(23,66,28,79),(24,75,29,62),(25,68,30,71),(31,76,36,63),(32,69,37,72),(33,78,38,65),(34,61,39,74),(35,80,40,67)], [(1,38,19,27),(2,34,20,23),(3,40,16,29),(4,36,17,25),(5,32,18,21),(6,33,13,22),(7,39,14,28),(8,35,15,24),(9,31,11,30),(10,37,12,26),(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,33),(22,32),(23,31),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(41,54),(42,53),(43,52),(44,51),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(61,63),(64,70),(65,69),(66,68),(71,79),(72,78),(73,77),(74,76)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 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⋊23D10 | C4×D20 | C4.D20 | D5×C22⋊C4 | C22⋊D20 | D10.12D4 | D10⋊D4 | Dic5.5D4 | D20⋊8C4 | D10.13D4 | C4⋊D20 | D10⋊Q8 | C4⋊C4⋊D5 | 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 | 4 | 2 | 6 | 6 | 2 | 4 | 8 |
Matrix representation of C42⋊23D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 1 | 37 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 1 | 7 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,11,32,0,0,0,0,9,30,0,0],[1,21,0,0,0,0,37,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,34,7] >;
C42⋊23D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{23}D_{10} % in TeX
G:=Group("C4^2:23D10"); // GroupNames label
G:=SmallGroup(320,1376);
// by ID
G=gap.SmallGroup(320,1376);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,192,12550]);
// 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=b^-1,d*c*d=c^-1>;
// generators/relations