metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊24D10, C10.1382+ (1+4), C4⋊C4⋊34D10, (C4×D20)⋊14C2, (C4×C20)⋊8C22, C42⋊2C2⋊3D5, C42⋊D5⋊6C2, 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, D10.12D4⋊49C2, D10.13D4⋊39C2, C2.63(D4⋊8D10), D10⋊C4⋊63C22, C23.55(C22×D5), Dic5.5D4⋊45C2, C5⋊9(C22.45C24), (C2×Dic10)⋊33C22, (C2×D20).234C22, C10.D4⋊68C22, C23.D10⋊45C2, (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
Subgroups: 1014 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×18], C5, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×5], C22×C4 [×5], C2×D4 [×3], C2×Q8, C24, Dic5 [×5], C20 [×6], D10 [×4], D10 [×11], C2×C10, C2×C10 [×3], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2, C42⋊2C2, Dic10, C4×D5 [×7], D20 [×4], C2×Dic5 [×5], C5⋊D4, C2×C20 [×6], C22×D5 [×3], C22×D5 [×5], C22×C10, C22.45C24, C4×Dic5 [×2], C10.D4 [×3], C4⋊Dic5 [×2], D10⋊C4 [×9], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×5], C2×D20 [×2], C2×C5⋊D4, C23×D5, C42⋊D5, C4×D20, C23.D10, D5×C22⋊C4 [×2], C22⋊D20, D10.12D4, Dic5.5D4, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4 [×2], D10⋊Q8, D10⋊2Q8, C5×C42⋊2C2, C42⋊24D10
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, D5×C4○D4 [×2], D4⋊8D10, C42⋊24D10
Generators and relations
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 >
►On 80 points
(1 53 8 47)(2 59 9 43)(3 55 10 49)(4 51 6 45)(5 57 7 41)(11 54 19 48)(12 60 20 44)(13 56 16 50)(14 52 17 46)(15 58 18 42)(21 70 26 73)(22 79 27 66)(23 62 28 75)(24 71 29 68)(25 64 30 77)(31 72 36 69)(32 65 37 78)(33 74 38 61)(34 67 39 80)(35 76 40 63)
(1 32 18 21)(2 38 19 27)(3 34 20 23)(4 40 16 29)(5 36 17 25)(6 35 13 24)(7 31 14 30)(8 37 15 26)(9 33 11 22)(10 39 12 28)(41 72 52 77)(42 70 53 65)(43 74 54 79)(44 62 55 67)(45 76 56 71)(46 64 57 69)(47 78 58 73)(48 66 59 61)(49 80 60 75)(50 68 51 63)
(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 9)(7 8)(11 13)(14 15)(16 19)(17 18)(21 31)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 42)(43 50)(44 49)(45 48)(46 47)(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)| (1,53,8,47)(2,59,9,43)(3,55,10,49)(4,51,6,45)(5,57,7,41)(11,54,19,48)(12,60,20,44)(13,56,16,50)(14,52,17,46)(15,58,18,42)(21,70,26,73)(22,79,27,66)(23,62,28,75)(24,71,29,68)(25,64,30,77)(31,72,36,69)(32,65,37,78)(33,74,38,61)(34,67,39,80)(35,76,40,63), (1,32,18,21)(2,38,19,27)(3,34,20,23)(4,40,16,29)(5,36,17,25)(6,35,13,24)(7,31,14,30)(8,37,15,26)(9,33,11,22)(10,39,12,28)(41,72,52,77)(42,70,53,65)(43,74,54,79)(44,62,55,67)(45,76,56,71)(46,64,57,69)(47,78,58,73)(48,66,59,61)(49,80,60,75)(50,68,51,63), (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)(6,9)(7,8)(11,13)(14,15)(16,19)(17,18)(21,31)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,42)(43,50)(44,49)(45,48)(46,47)(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( (1,53,8,47)(2,59,9,43)(3,55,10,49)(4,51,6,45)(5,57,7,41)(11,54,19,48)(12,60,20,44)(13,56,16,50)(14,52,17,46)(15,58,18,42)(21,70,26,73)(22,79,27,66)(23,62,28,75)(24,71,29,68)(25,64,30,77)(31,72,36,69)(32,65,37,78)(33,74,38,61)(34,67,39,80)(35,76,40,63), (1,32,18,21)(2,38,19,27)(3,34,20,23)(4,40,16,29)(5,36,17,25)(6,35,13,24)(7,31,14,30)(8,37,15,26)(9,33,11,22)(10,39,12,28)(41,72,52,77)(42,70,53,65)(43,74,54,79)(44,62,55,67)(45,76,56,71)(46,64,57,69)(47,78,58,73)(48,66,59,61)(49,80,60,75)(50,68,51,63), (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)(6,9)(7,8)(11,13)(14,15)(16,19)(17,18)(21,31)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,42)(43,50)(44,49)(45,48)(46,47)(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([(1,53,8,47),(2,59,9,43),(3,55,10,49),(4,51,6,45),(5,57,7,41),(11,54,19,48),(12,60,20,44),(13,56,16,50),(14,52,17,46),(15,58,18,42),(21,70,26,73),(22,79,27,66),(23,62,28,75),(24,71,29,68),(25,64,30,77),(31,72,36,69),(32,65,37,78),(33,74,38,61),(34,67,39,80),(35,76,40,63)], [(1,32,18,21),(2,38,19,27),(3,34,20,23),(4,40,16,29),(5,36,17,25),(6,35,13,24),(7,31,14,30),(8,37,15,26),(9,33,11,22),(10,39,12,28),(41,72,52,77),(42,70,53,65),(43,74,54,79),(44,62,55,67),(45,76,56,71),(46,64,57,69),(47,78,58,73),(48,66,59,61),(49,80,60,75),(50,68,51,63)], [(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),(6,9),(7,8),(11,13),(14,15),(16,19),(17,18),(21,31),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,42),(43,50),(44,49),(45,48),(46,47),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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