direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D10.C23, D10.13C24, C4⋊F5⋊5C22, (C4×F5)⋊9C22, (C22×C4)⋊10F5, C10⋊(C42⋊C2), (C22×C20)⋊14C4, C2.7(C23×F5), D5⋊(C42⋊C2), C10.6(C23×C4), (C2×F5).2C23, C23.54(C2×F5), C4.59(C22×F5), C20.84(C22×C4), (C4×D5).93C23, D10.28(C4○D4), C22⋊F5.6C22, (C22×Dic5)⋊22C4, D10.47(C22×C4), C22.20(C22×F5), Dic5.46(C22×C4), (C22×F5).19C22, (C23×D5).139C22, (C22×D5).284C23, (C2×C4×D5)⋊22C4, (C2×C4⋊F5)⋊9C2, C5⋊(C2×C42⋊C2), (C2×C4×F5)⋊10C2, (C2×C4)⋊12(C2×F5), (C2×C20)⋊13(C2×C4), (C4×D5)⋊20(C2×C4), D5.1(C2×C4○D4), (D5×C22×C4).33C2, (C2×C22⋊F5).9C2, (C2×Dic5)⋊37(C2×C4), (C2×C4×D5).406C22, (C22×C10).82(C2×C4), (C2×C10).100(C22×C4), (C22×D5).135(C2×C4), SmallGroup(320,1592)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D10.C23
G = < a,b,c,d,e,f | a2=b10=c2=f2=1, d2=b-1c, e2=b5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, dbd-1=b3, be=eb, bf=fb, dcd-1=b2c, ce=ec, cf=fc, de=ed, fdf=b5d, ef=fe >
Subgroups: 1098 in 330 conjugacy classes, 148 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, F5, D10, D10, D10, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C2×C42⋊C2, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C22×F5, C23×D5, C2×C4×F5, C2×C4⋊F5, D10.C23, C2×C22⋊F5, D5×C22×C4, C2×D10.C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, F5, C42⋊C2, C23×C4, C2×C4○D4, C2×F5, C2×C42⋊C2, C22×F5, D10.C23, C23×F5, C2×D10.C23
►On 80 points
Generators in S80
(1 18)(2 19)(3 20)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(10 17)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 31)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)
(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 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 60)(50 59)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 80)(70 79)
(1 59 13 41)(2 56 12 44)(3 53 11 47)(4 60 20 50)(5 57 19 43)(6 54 18 46)(7 51 17 49)(8 58 16 42)(9 55 15 45)(10 52 14 48)(21 77 39 63)(22 74 38 66)(23 71 37 69)(24 78 36 62)(25 75 35 65)(26 72 34 68)(27 79 33 61)(28 76 32 64)(29 73 31 67)(30 80 40 70)
(1 22 6 27)(2 23 7 28)(3 24 8 29)(4 25 9 30)(5 26 10 21)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 18)(2 19)(3 20)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(10 17)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 31)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(61 79)(62 80)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)
G:=sub<Sym(80)| (1,18)(2,19)(3,20)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,31)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73), (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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,60)(50,59)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,80)(70,79), (1,59,13,41)(2,56,12,44)(3,53,11,47)(4,60,20,50)(5,57,19,43)(6,54,18,46)(7,51,17,49)(8,58,16,42)(9,55,15,45)(10,52,14,48)(21,77,39,63)(22,74,38,66)(23,71,37,69)(24,78,36,62)(25,75,35,65)(26,72,34,68)(27,79,33,61)(28,76,32,64)(29,73,31,67)(30,80,40,70), (1,22,6,27)(2,23,7,28)(3,24,8,29)(4,25,9,30)(5,26,10,21)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,18)(2,19)(3,20)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,31)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)>;
G:=Group( (1,18)(2,19)(3,20)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,31)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73), (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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,60)(50,59)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,80)(70,79), (1,59,13,41)(2,56,12,44)(3,53,11,47)(4,60,20,50)(5,57,19,43)(6,54,18,46)(7,51,17,49)(8,58,16,42)(9,55,15,45)(10,52,14,48)(21,77,39,63)(22,74,38,66)(23,71,37,69)(24,78,36,62)(25,75,35,65)(26,72,34,68)(27,79,33,61)(28,76,32,64)(29,73,31,67)(30,80,40,70), (1,22,6,27)(2,23,7,28)(3,24,8,29)(4,25,9,30)(5,26,10,21)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,18)(2,19)(3,20)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,31)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78) );
G=PermutationGroup([[(1,18),(2,19),(3,20),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(10,17),(21,32),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,31),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,51),(49,52),(50,53),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73)], [(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)], [(1,12),(2,11),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,40),(30,39),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,60),(50,59),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,80),(70,79)], [(1,59,13,41),(2,56,12,44),(3,53,11,47),(4,60,20,50),(5,57,19,43),(6,54,18,46),(7,51,17,49),(8,58,16,42),(9,55,15,45),(10,52,14,48),(21,77,39,63),(22,74,38,66),(23,71,37,69),(24,78,36,62),(25,75,35,65),(26,72,34,68),(27,79,33,61),(28,76,32,64),(29,73,31,67),(30,80,40,70)], [(1,22,6,27),(2,23,7,28),(3,24,8,29),(4,25,9,30),(5,26,10,21),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(1,18),(2,19),(3,20),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(10,17),(21,32),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,31),(41,59),(42,60),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(49,57),(50,58),(61,79),(62,80),(63,71),(64,72),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4AB | 5 | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | F5 | C2×F5 | C2×F5 | D10.C23 |
| kernel | C2×D10.C23 | C2×C4×F5 | C2×C4⋊F5 | D10.C23 | C2×C22⋊F5 | D5×C22×C4 | C2×C4×D5 | C22×Dic5 | C22×C20 | D10 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 2 | 1 | 12 | 2 | 2 | 8 | 1 | 6 | 1 | 8 |
Matrix representation of C2×D10.C23 ►in GL6(𝔽41)
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 18 | 30 | 0 | 0 | 0 | 0 |
| 37 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 28 | 28 |
| 0 | 0 | 28 | 28 | 0 | 8 |
| 0 | 0 | 13 | 21 | 13 | 0 |
| 0 | 0 | 33 | 20 | 20 | 33 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 9 | 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 |
G:=sub<GL(6,GF(41))| [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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,40,0,0,1,0,0,0,0,0,1],[18,37,0,0,0,0,30,23,0,0,0,0,0,0,8,28,13,33,0,0,0,28,21,20,0,0,28,0,13,20,0,0,28,8,0,33],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,9,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
C2×D10.C23 in GAP, Magma, Sage, TeX
C_2\times D_{10}.C_2^3 % in TeX
G:=Group("C2xD10.C2^3"); // GroupNames label
G:=SmallGroup(320,1592);
// by ID
G=gap.SmallGroup(320,1592);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1123,6278,818]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^10=c^2=f^2=1,d^2=b^-1*c,e^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,d*b*d^-1=b^3,b*e=e*b,b*f=f*b,d*c*d^-1=b^2*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=b^5*d,e*f=f*e>;
// generators/relations