metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.35D10, C10.292+ 1+4, C22≀C2⋊6D5, C20⋊2D4⋊14C2, (D4×Dic5)⋊13C2, (C2×D4).86D10, C24⋊2D5⋊8C2, C22⋊C4.2D10, Dic5⋊4D4⋊4C2, Dic5⋊D4⋊5C2, (C2×C20).31C23, C4⋊Dic5⋊27C22, C20.17D4⋊12C2, (C2×C10).137C24, C5⋊3(C22.32C24), (C4×Dic5)⋊17C22, D10.12D4⋊14C2, C23.D5⋊17C22, C2.31(D4⋊6D10), D10⋊C4⋊14C22, Dic5.5D4⋊14C2, (C2×Dic10)⋊22C22, (D4×C10).111C22, C10.D4⋊12C22, C23.18D10⋊5C2, C23.D10⋊12C2, (C23×C10).70C22, (C2×Dic5).62C23, (C22×D5).56C23, C22.158(C23×D5), C23.177(C22×D5), Dic5.14D4⋊14C2, C22.10(D4⋊2D5), (C22×C10).182C23, (C22×Dic5)⋊16C22, (C2×C4×D5)⋊10C22, (C5×C22≀C2)⋊8C2, C10.78(C2×C4○D4), C2.29(C2×D4⋊2D5), (C2×C5⋊D4)⋊10C22, (C2×C23.D5)⋊21C2, (C2×C4).31(C22×D5), (C2×C10).44(C4○D4), (C5×C22⋊C4).3C22, SmallGroup(320,1265)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.35D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 878 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22.32C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×C10, C23×C10, Dic5.14D4, C23.D10, Dic5⋊4D4, D10.12D4, Dic5.5D4, D4×Dic5, C23.18D10, C20.17D4, C20⋊2D4, Dic5⋊D4, C2×C23.D5, C24⋊2D5, C5×C22≀C2, C24.35D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, D4⋊6D10, C24.35D10
►On 80 points
(2 25)(4 27)(6 29)(8 31)(10 33)(12 35)(14 37)(16 39)(18 21)(20 23)(41 51)(42 63)(43 53)(44 65)(45 55)(46 67)(47 57)(48 69)(49 59)(50 71)(52 73)(54 75)(56 77)(58 79)(60 61)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 11)(3 13)(5 15)(7 17)(9 19)(22 32)(24 34)(26 36)(28 38)(30 40)(41 62)(42 73)(43 64)(44 75)(45 66)(46 77)(47 68)(48 79)(49 70)(50 61)(51 72)(52 63)(53 74)(54 65)(55 76)(56 67)(57 78)(58 69)(59 80)(60 71)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(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 54 11 44)(2 43 12 53)(3 52 13 42)(4 41 14 51)(5 50 15 60)(6 59 16 49)(7 48 17 58)(8 57 18 47)(9 46 19 56)(10 55 20 45)(21 78 31 68)(22 67 32 77)(23 76 33 66)(24 65 34 75)(25 74 35 64)(26 63 36 73)(27 72 37 62)(28 61 38 71)(29 70 39 80)(30 79 40 69)
G:=sub<Sym(80)| (2,25)(4,27)(6,29)(8,31)(10,33)(12,35)(14,37)(16,39)(18,21)(20,23)(41,51)(42,63)(43,53)(44,65)(45,55)(46,67)(47,57)(48,69)(49,59)(50,71)(52,73)(54,75)(56,77)(58,79)(60,61)(62,72)(64,74)(66,76)(68,78)(70,80), (1,11)(3,13)(5,15)(7,17)(9,19)(22,32)(24,34)(26,36)(28,38)(30,40)(41,62)(42,73)(43,64)(44,75)(45,66)(46,77)(47,68)(48,79)(49,70)(50,61)(51,72)(52,63)(53,74)(54,65)(55,76)(56,67)(57,78)(58,69)(59,80)(60,71), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,54,11,44)(2,43,12,53)(3,52,13,42)(4,41,14,51)(5,50,15,60)(6,59,16,49)(7,48,17,58)(8,57,18,47)(9,46,19,56)(10,55,20,45)(21,78,31,68)(22,67,32,77)(23,76,33,66)(24,65,34,75)(25,74,35,64)(26,63,36,73)(27,72,37,62)(28,61,38,71)(29,70,39,80)(30,79,40,69)>;
G:=Group( (2,25)(4,27)(6,29)(8,31)(10,33)(12,35)(14,37)(16,39)(18,21)(20,23)(41,51)(42,63)(43,53)(44,65)(45,55)(46,67)(47,57)(48,69)(49,59)(50,71)(52,73)(54,75)(56,77)(58,79)(60,61)(62,72)(64,74)(66,76)(68,78)(70,80), (1,11)(3,13)(5,15)(7,17)(9,19)(22,32)(24,34)(26,36)(28,38)(30,40)(41,62)(42,73)(43,64)(44,75)(45,66)(46,77)(47,68)(48,79)(49,70)(50,61)(51,72)(52,63)(53,74)(54,65)(55,76)(56,67)(57,78)(58,69)(59,80)(60,71), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,54,11,44)(2,43,12,53)(3,52,13,42)(4,41,14,51)(5,50,15,60)(6,59,16,49)(7,48,17,58)(8,57,18,47)(9,46,19,56)(10,55,20,45)(21,78,31,68)(22,67,32,77)(23,76,33,66)(24,65,34,75)(25,74,35,64)(26,63,36,73)(27,72,37,62)(28,61,38,71)(29,70,39,80)(30,79,40,69) );
G=PermutationGroup([[(2,25),(4,27),(6,29),(8,31),(10,33),(12,35),(14,37),(16,39),(18,21),(20,23),(41,51),(42,63),(43,53),(44,65),(45,55),(46,67),(47,57),(48,69),(49,59),(50,71),(52,73),(54,75),(56,77),(58,79),(60,61),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,11),(3,13),(5,15),(7,17),(9,19),(22,32),(24,34),(26,36),(28,38),(30,40),(41,62),(42,73),(43,64),(44,75),(45,66),(46,77),(47,68),(48,79),(49,70),(50,61),(51,72),(52,63),(53,74),(54,65),(55,76),(56,67),(57,78),(58,69),(59,80),(60,71)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,21),(19,22),(20,23),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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)], [(1,54,11,44),(2,43,12,53),(3,52,13,42),(4,41,14,51),(5,50,15,60),(6,59,16,49),(7,48,17,58),(8,57,18,47),(9,46,19,56),(10,55,20,45),(21,78,31,68),(22,67,32,77),(23,76,33,66),(24,65,34,75),(25,74,35,64),(26,63,36,73),(27,72,37,62),(28,61,38,71),(29,70,39,80),(30,79,40,69)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| 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 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 20 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 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 | D4⋊2D5 | D4⋊6D10 |
| kernel | C24.35D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | D4×Dic5 | C23.18D10 | C20.17D4 | C20⋊2D4 | Dic5⋊D4 | C2×C23.D5 | C24⋊2D5 | C5×C22≀C2 | C22≀C2 | C2×C10 | C22⋊C4 | C2×D4 | C24 | C10 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 6 | 6 | 2 | 2 | 4 | 8 |
Matrix representation of C24.35D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,37,0,0,0,0,37,0,0,0,0,10,0,0,0,0,10,0,0,0] >;
C24.35D10 in GAP, Magma, Sage, TeX
C_2^4._{35}D_{10} % in TeX
G:=Group("C2^4.35D10"); // GroupNames label
G:=SmallGroup(320,1265);
// by ID
G=gap.SmallGroup(320,1265);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations