metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊21D10, C10.732+ (1+4), (C2×Q8)⋊9D10, (C4×C20)⋊33C22, C22⋊C4⋊35D10, C4.4D4⋊13D5, D10⋊3Q8⋊31C2, C23⋊D10.6C2, (C2×D4).111D10, C42⋊D5⋊36C2, C4⋊Dic5⋊42C22, (Q8×C10)⋊15C22, D10.37(C4○D4), Dic5⋊4D4⋊32C2, (C2×C10).223C24, (C2×C20).632C23, (C4×Dic5)⋊57C22, D10.12D4⋊44C2, C2.76(D4⋊6D10), C23.D5⋊34C22, C23.45(C22×D5), C5⋊8(C22.45C24), (D4×C10).211C22, C10.D4⋊67C22, C23.D10⋊40C2, (C22×C10).53C23, (C23×D5).66C22, C22.244(C23×D5), C23.18D10⋊25C2, (C2×Dic5).265C23, (C22×Dic5)⋊28C22, (C22×D5).227C23, D10⋊C4.136C22, C2.79(D5×C4○D4), (D5×C22⋊C4)⋊19C2, C10.190(C2×C4○D4), (C5×C4.4D4)⋊15C2, (C2×C4×D5).266C22, (C2×C4).74(C22×D5), (C5×C22⋊C4)⋊31C22, (C2×C5⋊D4).61C22, SmallGroup(320,1351)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 950 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×18], C5, C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D5 [×4], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×6], C20 [×5], D10 [×4], D10 [×8], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], C4×D5 [×6], C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4, C5×Q8, C22×D5 [×2], C22×D5 [×5], C22×C10 [×2], C22.45C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4×C10, Q8×C10, C23×D5, C42⋊D5 [×2], C23.D10 [×2], D5×C22⋊C4 [×2], Dic5⋊4D4 [×2], D10.12D4 [×2], C23.18D10, C23⋊D10, D10⋊3Q8 [×2], C5×C4.4D4, C42⋊21D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.45C24, C23×D5, D4⋊6D10, D5×C4○D4 [×2], C42⋊21D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 80 points
(1 53 12 43)(2 59 13 49)(3 55 14 45)(4 51 15 41)(5 57 11 47)(6 56 16 46)(7 52 17 42)(8 58 18 48)(9 54 19 44)(10 60 20 50)(21 65 26 73)(22 79 27 61)(23 67 28 75)(24 71 29 63)(25 69 30 77)(31 70 36 78)(32 74 37 66)(33 62 38 80)(34 76 39 68)(35 64 40 72)
(1 36 8 21)(2 32 9 27)(3 38 10 23)(4 34 6 29)(5 40 7 25)(11 35 17 30)(12 31 18 26)(13 37 19 22)(14 33 20 28)(15 39 16 24)(41 68 46 71)(42 77 47 64)(43 70 48 73)(44 79 49 66)(45 62 50 75)(51 76 56 63)(52 69 57 72)(53 78 58 65)(54 61 59 74)(55 80 60 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 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)(21 25)(22 24)(26 30)(27 29)(31 35)(32 34)(36 40)(37 39)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(50 60)(61 63)(64 70)(65 69)(66 68)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,53,12,43)(2,59,13,49)(3,55,14,45)(4,51,15,41)(5,57,11,47)(6,56,16,46)(7,52,17,42)(8,58,18,48)(9,54,19,44)(10,60,20,50)(21,65,26,73)(22,79,27,61)(23,67,28,75)(24,71,29,63)(25,69,30,77)(31,70,36,78)(32,74,37,66)(33,62,38,80)(34,76,39,68)(35,64,40,72), (1,36,8,21)(2,32,9,27)(3,38,10,23)(4,34,6,29)(5,40,7,25)(11,35,17,30)(12,31,18,26)(13,37,19,22)(14,33,20,28)(15,39,16,24)(41,68,46,71)(42,77,47,64)(43,70,48,73)(44,79,49,66)(45,62,50,75)(51,76,56,63)(52,69,57,72)(53,78,58,65)(54,61,59,74)(55,80,60,67), (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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)(36,40)(37,39)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,63)(64,70)(65,69)(66,68)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,53,12,43)(2,59,13,49)(3,55,14,45)(4,51,15,41)(5,57,11,47)(6,56,16,46)(7,52,17,42)(8,58,18,48)(9,54,19,44)(10,60,20,50)(21,65,26,73)(22,79,27,61)(23,67,28,75)(24,71,29,63)(25,69,30,77)(31,70,36,78)(32,74,37,66)(33,62,38,80)(34,76,39,68)(35,64,40,72), (1,36,8,21)(2,32,9,27)(3,38,10,23)(4,34,6,29)(5,40,7,25)(11,35,17,30)(12,31,18,26)(13,37,19,22)(14,33,20,28)(15,39,16,24)(41,68,46,71)(42,77,47,64)(43,70,48,73)(44,79,49,66)(45,62,50,75)(51,76,56,63)(52,69,57,72)(53,78,58,65)(54,61,59,74)(55,80,60,67), (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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)(36,40)(37,39)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,63)(64,70)(65,69)(66,68)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([(1,53,12,43),(2,59,13,49),(3,55,14,45),(4,51,15,41),(5,57,11,47),(6,56,16,46),(7,52,17,42),(8,58,18,48),(9,54,19,44),(10,60,20,50),(21,65,26,73),(22,79,27,61),(23,67,28,75),(24,71,29,63),(25,69,30,77),(31,70,36,78),(32,74,37,66),(33,62,38,80),(34,76,39,68),(35,64,40,72)], [(1,36,8,21),(2,32,9,27),(3,38,10,23),(4,34,6,29),(5,40,7,25),(11,35,17,30),(12,31,18,26),(13,37,19,22),(14,33,20,28),(15,39,16,24),(41,68,46,71),(42,77,47,64),(43,70,48,73),(44,79,49,66),(45,62,50,75),(51,76,56,63),(52,69,57,72),(53,78,58,65),(54,61,59,74),(55,80,60,67)], [(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)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20),(21,25),(22,24),(26,30),(27,29),(31,35),(32,34),(36,40),(37,39),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(50,60),(61,63),(64,70),(65,69),(66,68),(71,79),(72,78),(73,77),(74,76)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
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 | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| 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 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 |
| kernel | C42⋊21D10 | C42⋊D5 | C23.D10 | D5×C22⋊C4 | Dic5⋊4D4 | D10.12D4 | C23.18D10 | C23⋊D10 | D10⋊3Q8 | C5×C4.4D4 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 8 | 2 | 8 | 2 | 2 | 1 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{21}D_{10} % in TeX
G:=Group("C4^2:21D10"); // GroupNames label
G:=SmallGroup(320,1351);
// by ID
G=gap.SmallGroup(320,1351);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,387,100,346,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations