metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊23D10, C10.1372+ (1+4), C4⋊C4⋊33D10, (C4×D20)⋊13C2, (C4×C20)⋊7C22, C4⋊2D20⋊36C2, C4.D20⋊8C2, C42⋊2C2⋊2D5, C22⋊D20⋊27C2, D20⋊8C4⋊39C2, 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, D10.12D4⋊48C2, D10.13D4⋊38C2, 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
Subgroups: 1134 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C5, C2×C4 [×6], C2×C4 [×8], D4 [×9], Q8, C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C42, C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×7], C2×Q8, C24, Dic5 [×4], C20 [×6], D10 [×2], D10 [×15], C2×C10, C2×C10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2, C42⋊2C2, Dic10, C4×D5 [×4], D20 [×7], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×6], C22×D5 [×4], C22×D5 [×4], C22×C10, C22.32C24, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×10], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×4], C2×D20 [×5], C2×C5⋊D4 [×2], C23×D5, C4×D20, C4.D20, D5×C22⋊C4, C22⋊D20 [×2], D10.12D4, D10⋊D4, Dic5.5D4, D20⋊8C4, D10.13D4, C4⋊2D20 [×2], D10⋊Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42⋊23D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D5 [×7], C22.32C24, C23×D5, D5×C4○D4, D4⋊8D10 [×2], C42⋊23D10
Generators and relations
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 >
►On 80 points
(1 57 7 43)(2 53 8 49)(3 59 9 45)(4 55 10 41)(5 51 6 47)(11 56 16 42)(12 52 17 48)(13 58 18 44)(14 54 19 50)(15 60 20 46)(21 66 26 80)(22 76 27 62)(23 68 28 72)(24 78 29 64)(25 70 30 74)(31 73 36 69)(32 65 37 79)(33 75 38 61)(34 67 39 71)(35 77 40 63)
(1 32 17 25)(2 38 18 21)(3 34 19 27)(4 40 20 23)(5 36 16 29)(6 31 11 24)(7 37 12 30)(8 33 13 26)(9 39 14 22)(10 35 15 28)(41 77 60 72)(42 64 51 69)(43 79 52 74)(44 66 53 61)(45 71 54 76)(46 68 55 63)(47 73 56 78)(48 70 57 65)(49 75 58 80)(50 62 59 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 8)(2 7)(3 6)(4 10)(5 9)(11 19)(12 18)(13 17)(14 16)(15 20)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 70)(67 69)(71 73)(74 80)(75 79)(76 78)
G:=sub<Sym(80)| (1,57,7,43)(2,53,8,49)(3,59,9,45)(4,55,10,41)(5,51,6,47)(11,56,16,42)(12,52,17,48)(13,58,18,44)(14,54,19,50)(15,60,20,46)(21,66,26,80)(22,76,27,62)(23,68,28,72)(24,78,29,64)(25,70,30,74)(31,73,36,69)(32,65,37,79)(33,75,38,61)(34,67,39,71)(35,77,40,63), (1,32,17,25)(2,38,18,21)(3,34,19,27)(4,40,20,23)(5,36,16,29)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(41,77,60,72)(42,64,51,69)(43,79,52,74)(44,66,53,61)(45,71,54,76)(46,68,55,63)(47,73,56,78)(48,70,57,65)(49,75,58,80)(50,62,59,67), (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,19)(12,18)(13,17)(14,16)(15,20)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,70)(67,69)(71,73)(74,80)(75,79)(76,78)>;
G:=Group( (1,57,7,43)(2,53,8,49)(3,59,9,45)(4,55,10,41)(5,51,6,47)(11,56,16,42)(12,52,17,48)(13,58,18,44)(14,54,19,50)(15,60,20,46)(21,66,26,80)(22,76,27,62)(23,68,28,72)(24,78,29,64)(25,70,30,74)(31,73,36,69)(32,65,37,79)(33,75,38,61)(34,67,39,71)(35,77,40,63), (1,32,17,25)(2,38,18,21)(3,34,19,27)(4,40,20,23)(5,36,16,29)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(41,77,60,72)(42,64,51,69)(43,79,52,74)(44,66,53,61)(45,71,54,76)(46,68,55,63)(47,73,56,78)(48,70,57,65)(49,75,58,80)(50,62,59,67), (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,19)(12,18)(13,17)(14,16)(15,20)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,70)(67,69)(71,73)(74,80)(75,79)(76,78) );
G=PermutationGroup([(1,57,7,43),(2,53,8,49),(3,59,9,45),(4,55,10,41),(5,51,6,47),(11,56,16,42),(12,52,17,48),(13,58,18,44),(14,54,19,50),(15,60,20,46),(21,66,26,80),(22,76,27,62),(23,68,28,72),(24,78,29,64),(25,70,30,74),(31,73,36,69),(32,65,37,79),(33,75,38,61),(34,67,39,71),(35,77,40,63)], [(1,32,17,25),(2,38,18,21),(3,34,19,27),(4,40,20,23),(5,36,16,29),(6,31,11,24),(7,37,12,30),(8,33,13,26),(9,39,14,22),(10,35,15,28),(41,77,60,72),(42,64,51,69),(43,79,52,74),(44,66,53,61),(45,71,54,76),(46,68,55,63),(47,73,56,78),(48,70,57,65),(49,75,58,80),(50,62,59,67)], [(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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,19),(12,18),(13,17),(14,16),(15,20),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,40),(29,39),(30,38),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,65),(62,64),(66,70),(67,69),(71,73),(74,80),(75,79),(76,78)])
Matrix representation ►G ⊆ 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] >;
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⋊2D20 | 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 |
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