metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊25D10, C10.1422+ (1+4), C4⋊C4⋊17D10, C4⋊D20⋊5C2, (C4×C20)⋊2C22, C4⋊2D20⋊38C2, C42⋊2D5⋊1C2, C42⋊2C2⋊8D5, C22⋊D20⋊28C2, D10⋊D4⋊46C2, (C2×D20)⋊10C22, C22⋊C4.41D10, (C2×C20).195C23, (C2×C10).255C24, C10.D4⋊5C22, D10.13D4⋊44C2, C2.67(D4⋊8D10), D10⋊C4⋊24C22, C23.61(C22×D5), C5⋊4(C22.54C24), (C22×C10).69C23, (C23×D5).70C22, C22.276(C23×D5), (C2×Dic5).131C23, (C22×D5).114C23, (C2×C4×D5)⋊28C22, (C5×C4⋊C4)⋊34C22, (C5×C42⋊2C2)⋊10C2, (C2×C4).211(C22×D5), (C2×C5⋊D4).75C22, (C5×C22⋊C4).80C22, SmallGroup(320,1383)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1254 in 252 conjugacy classes, 91 normal (16 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4 [×6], C2×C4 [×6], D4 [×12], C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×3], C2×D4 [×12], C24, Dic5 [×3], C20 [×6], D10 [×19], C2×C10, C2×C10 [×3], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2, C42⋊2C2, C4⋊1D4, C4×D5 [×3], D20 [×9], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×6], C22×D5 [×2], C22×D5 [×3], C22×D5 [×3], C22×C10, C22.54C24, C10.D4 [×3], D10⋊C4 [×9], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×3], C2×D20 [×9], C2×C5⋊D4 [×3], C23×D5, C4⋊D20, C42⋊2D5, C22⋊D20 [×3], D10⋊D4 [×3], D10.13D4 [×3], C4⋊2D20 [×3], C5×C42⋊2C2, C42⋊25D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4) [×3], C22×D5 [×7], C22.54C24, C23×D5, D4⋊8D10 [×3], C42⋊25D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
►On 80 points
(1 51 6 41)(2 57 7 47)(3 53 8 43)(4 59 9 49)(5 55 10 45)(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 76 36 63)(32 69 37 72)(33 78 38 65)(34 61 39 74)(35 80 40 67)
(1 36 16 29)(2 32 17 25)(3 38 18 21)(4 34 19 27)(5 40 20 23)(6 31 11 24)(7 37 12 30)(8 33 13 26)(9 39 14 22)(10 35 15 28)(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 9)(2 8)(3 7)(4 6)(5 10)(11 19)(12 18)(13 17)(14 16)(15 20)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(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)| (1,51,6,41)(2,57,7,47)(3,53,8,43)(4,59,9,49)(5,55,10,45)(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)(31,76,36,63)(32,69,37,72)(33,78,38,65)(34,61,39,74)(35,80,40,67), (1,36,16,29)(2,32,17,25)(3,38,18,21)(4,34,19,27)(5,40,20,23)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(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), (1,9)(2,8)(3,7)(4,6)(5,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(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( (1,51,6,41)(2,57,7,47)(3,53,8,43)(4,59,9,49)(5,55,10,45)(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)(31,76,36,63)(32,69,37,72)(33,78,38,65)(34,61,39,74)(35,80,40,67), (1,36,16,29)(2,32,17,25)(3,38,18,21)(4,34,19,27)(5,40,20,23)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(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), (1,9)(2,8)(3,7)(4,6)(5,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(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([(1,51,6,41),(2,57,7,47),(3,53,8,43),(4,59,9,49),(5,55,10,45),(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),(31,76,36,63),(32,69,37,72),(33,78,38,65),(34,61,39,74),(35,80,40,67)], [(1,36,16,29),(2,32,17,25),(3,38,18,21),(4,34,19,27),(5,40,20,23),(6,31,11,24),(7,37,12,30),(8,33,13,26),(9,39,14,22),(10,35,15,28),(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)], [(1,9),(2,8),(3,7),(4,6),(5,10),(11,19),(12,18),(13,17),(14,16),(15,20),(21,32),(22,31),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(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)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 2 | 28 | 10 | 12 | 0 | 0 | 0 | 0 |
| 13 | 39 | 10 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 32 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 30 |
| 1 | 0 | 28 | 24 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 32 | 39 | 0 |
| 0 | 0 | 0 | 0 | 9 | 30 | 0 | 39 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 11 |
| 40 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 14 | 6 | 6 | 0 | 0 | 0 | 0 |
| 14 | 32 | 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 14 | 1 | 34 |
| 0 | 0 | 0 | 0 | 27 | 27 | 7 | 34 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 27 | 35 | 35 | 0 | 0 | 0 | 0 |
| 27 | 30 | 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 32 | 40 | 0 |
| 0 | 0 | 0 | 0 | 27 | 30 | 34 | 1 |
G:=sub<GL(8,GF(41))| [2,13,0,0,0,0,0,0,28,39,0,0,0,0,0,0,10,10,2,28,0,0,0,0,12,10,13,39,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,13,40,0,0,0,0,0,24,28,0,40,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30,0,0,0,0,0,0,39,0,30,32,0,0,0,0,0,39,9,11],[40,6,32,14,0,0,0,0,35,35,14,32,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,40,34,11,27,0,0,0,0,7,7,14,27,0,0,0,0,0,0,1,7,0,0,0,0,0,0,34,34],[1,35,9,27,0,0,0,0,0,40,27,30,0,0,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,0,0,0,0,40,34,11,27,0,0,0,0,0,1,32,30,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,1] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | ··· | 20 | 4 | ··· | 4 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | 2+ (1+4) | D4⋊8D10 |
| kernel | C42⋊25D10 | C4⋊D20 | C42⋊2D5 | C22⋊D20 | D10⋊D4 | D10.13D4 | C4⋊2D20 | C5×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 |
| # reps | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 2 | 2 | 6 | 6 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{25}D_{10} % in TeX
G:=Group("C4^2:25D10"); // GroupNames label
G:=SmallGroup(320,1383);
// by ID
G=gap.SmallGroup(320,1383);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,184,1571,570,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations