metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊25D10, C10.1422+ 1+4, C4⋊C4⋊17D10, C20⋊4D4⋊5C2, (C4×C20)⋊2C22, C4⋊D20⋊38C2, C42⋊2C2⋊8D5, C42⋊2D5⋊1C2, D10⋊D4⋊46C2, C22⋊D20⋊28C2, (C2×D20)⋊10C22, C22⋊C4.41D10, (C2×C10).255C24, (C2×C20).195C23, 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
Generators and relations for C42⋊25D10
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 >
Subgroups: 1254 in 252 conjugacy classes, 91 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C42⋊2C2, C4⋊1D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, C22.54C24, C10.D4, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C23×D5, C20⋊4D4, C42⋊2D5, C22⋊D20, D10⋊D4, D10.13D4, C4⋊D20, C5×C42⋊2C2, C42⋊25D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C22.54C24, C23×D5, D4⋊8D10, C42⋊25D10
►On 80 points
(1 67 6 46)(2 63 7 42)(3 69 8 48)(4 65 9 44)(5 61 10 50)(11 64 19 43)(12 70 20 49)(13 66 16 45)(14 62 17 41)(15 68 18 47)(21 78 26 57)(22 53 27 74)(23 80 28 59)(24 55 29 76)(25 72 30 51)(31 60 36 71)(32 77 37 56)(33 52 38 73)(34 79 39 58)(35 54 40 75)
(1 40 17 23)(2 36 18 29)(3 32 19 25)(4 38 20 21)(5 34 16 27)(6 35 14 28)(7 31 15 24)(8 37 11 30)(9 33 12 26)(10 39 13 22)(41 80 67 75)(42 60 68 55)(43 72 69 77)(44 52 70 57)(45 74 61 79)(46 54 62 59)(47 76 63 71)(48 56 64 51)(49 78 65 73)(50 58 66 53)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 20)(12 19)(13 18)(14 17)(15 16)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(41 46)(42 45)(43 44)(47 50)(48 49)(51 57)(52 56)(53 55)(58 60)(61 68)(62 67)(63 66)(64 65)(69 70)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,67,6,46)(2,63,7,42)(3,69,8,48)(4,65,9,44)(5,61,10,50)(11,64,19,43)(12,70,20,49)(13,66,16,45)(14,62,17,41)(15,68,18,47)(21,78,26,57)(22,53,27,74)(23,80,28,59)(24,55,29,76)(25,72,30,51)(31,60,36,71)(32,77,37,56)(33,52,38,73)(34,79,39,58)(35,54,40,75), (1,40,17,23)(2,36,18,29)(3,32,19,25)(4,38,20,21)(5,34,16,27)(6,35,14,28)(7,31,15,24)(8,37,11,30)(9,33,12,26)(10,39,13,22)(41,80,67,75)(42,60,68,55)(43,72,69,77)(44,52,70,57)(45,74,61,79)(46,54,62,59)(47,76,63,71)(48,56,64,51)(49,78,65,73)(50,58,66,53), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,20)(12,19)(13,18)(14,17)(15,16)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(41,46)(42,45)(43,44)(47,50)(48,49)(51,57)(52,56)(53,55)(58,60)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,67,6,46)(2,63,7,42)(3,69,8,48)(4,65,9,44)(5,61,10,50)(11,64,19,43)(12,70,20,49)(13,66,16,45)(14,62,17,41)(15,68,18,47)(21,78,26,57)(22,53,27,74)(23,80,28,59)(24,55,29,76)(25,72,30,51)(31,60,36,71)(32,77,37,56)(33,52,38,73)(34,79,39,58)(35,54,40,75), (1,40,17,23)(2,36,18,29)(3,32,19,25)(4,38,20,21)(5,34,16,27)(6,35,14,28)(7,31,15,24)(8,37,11,30)(9,33,12,26)(10,39,13,22)(41,80,67,75)(42,60,68,55)(43,72,69,77)(44,52,70,57)(45,74,61,79)(46,54,62,59)(47,76,63,71)(48,56,64,51)(49,78,65,73)(50,58,66,53), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,20)(12,19)(13,18)(14,17)(15,16)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(41,46)(42,45)(43,44)(47,50)(48,49)(51,57)(52,56)(53,55)(58,60)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,67,6,46),(2,63,7,42),(3,69,8,48),(4,65,9,44),(5,61,10,50),(11,64,19,43),(12,70,20,49),(13,66,16,45),(14,62,17,41),(15,68,18,47),(21,78,26,57),(22,53,27,74),(23,80,28,59),(24,55,29,76),(25,72,30,51),(31,60,36,71),(32,77,37,56),(33,52,38,73),(34,79,39,58),(35,54,40,75)], [(1,40,17,23),(2,36,18,29),(3,32,19,25),(4,38,20,21),(5,34,16,27),(6,35,14,28),(7,31,15,24),(8,37,11,30),(9,33,12,26),(10,39,13,22),(41,80,67,75),(42,60,68,55),(43,72,69,77),(44,52,70,57),(45,74,61,79),(46,54,62,59),(47,76,63,71),(48,56,64,51),(49,78,65,73),(50,58,66,53)], [(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)], [(1,6),(2,10),(3,9),(4,8),(5,7),(11,20),(12,19),(13,18),(14,17),(15,16),(21,32),(22,31),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(41,46),(42,45),(43,44),(47,50),(48,49),(51,57),(52,56),(53,55),(58,60),(61,68),(62,67),(63,66),(64,65),(69,70),(71,79),(72,78),(73,77),(74,76)]])
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 | C20⋊4D4 | C42⋊2D5 | C22⋊D20 | D10⋊D4 | D10.13D4 | C4⋊D20 | 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 |
Matrix representation of C42⋊25D10 ►in 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] >;
C42⋊25D10 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