metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.3D10, C4⋊Dic5⋊3C2, (C2×C4).27D10, C22⋊C4.2D5, C5⋊2(C42⋊2C2), (C4×Dic5)⋊10C2, C10.7(C4○D4), C2.9(C4○D20), C10.D4⋊8C2, (C2×C20).2C22, C23.D5.3C2, C2.7(D4⋊2D5), (C2×C10).20C23, (C22×C10).9C22, C22.40(C22×D5), (C2×Dic5).29C22, (C5×C22⋊C4).2C2, SmallGroup(160,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D10
G = < a,b,c,d,e | a2=b2=c2=1, d10=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >
Subgroups: 160 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C42⋊2C2, C2×Dic5, C2×C20, C22×C10, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C23.D10
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C42⋊2C2, C22×D5, C4○D20, D4⋊2D5, C23.D10
►On 80 points
(2 70)(4 72)(6 74)(8 76)(10 78)(12 80)(14 62)(16 64)(18 66)(20 68)(21 31)(22 52)(23 33)(24 54)(25 35)(26 56)(27 37)(28 58)(29 39)(30 60)(32 42)(34 44)(36 46)(38 48)(40 50)(41 51)(43 53)(45 55)(47 57)(49 59)
(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 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)
(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 35 79 45)(2 24 80 54)(3 33 61 43)(4 22 62 52)(5 31 63 41)(6 40 64 50)(7 29 65 59)(8 38 66 48)(9 27 67 57)(10 36 68 46)(11 25 69 55)(12 34 70 44)(13 23 71 53)(14 32 72 42)(15 21 73 51)(16 30 74 60)(17 39 75 49)(18 28 76 58)(19 37 77 47)(20 26 78 56)
G:=sub<Sym(80)| (2,70)(4,72)(6,74)(8,76)(10,78)(12,80)(14,62)(16,64)(18,66)(20,68)(21,31)(22,52)(23,33)(24,54)(25,35)(26,56)(27,37)(28,58)(29,39)(30,60)(32,42)(34,44)(36,46)(38,48)(40,50)(41,51)(43,53)(45,55)(47,57)(49,59), (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), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60), (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), (1,35,79,45)(2,24,80,54)(3,33,61,43)(4,22,62,52)(5,31,63,41)(6,40,64,50)(7,29,65,59)(8,38,66,48)(9,27,67,57)(10,36,68,46)(11,25,69,55)(12,34,70,44)(13,23,71,53)(14,32,72,42)(15,21,73,51)(16,30,74,60)(17,39,75,49)(18,28,76,58)(19,37,77,47)(20,26,78,56)>;
G:=Group( (2,70)(4,72)(6,74)(8,76)(10,78)(12,80)(14,62)(16,64)(18,66)(20,68)(21,31)(22,52)(23,33)(24,54)(25,35)(26,56)(27,37)(28,58)(29,39)(30,60)(32,42)(34,44)(36,46)(38,48)(40,50)(41,51)(43,53)(45,55)(47,57)(49,59), (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), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60), (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), (1,35,79,45)(2,24,80,54)(3,33,61,43)(4,22,62,52)(5,31,63,41)(6,40,64,50)(7,29,65,59)(8,38,66,48)(9,27,67,57)(10,36,68,46)(11,25,69,55)(12,34,70,44)(13,23,71,53)(14,32,72,42)(15,21,73,51)(16,30,74,60)(17,39,75,49)(18,28,76,58)(19,37,77,47)(20,26,78,56) );
G=PermutationGroup([[(2,70),(4,72),(6,74),(8,76),(10,78),(12,80),(14,62),(16,64),(18,66),(20,68),(21,31),(22,52),(23,33),(24,54),(25,35),(26,56),(27,37),(28,58),(29,39),(30,60),(32,42),(34,44),(36,46),(38,48),(40,50),(41,51),(43,53),(45,55),(47,57),(49,59)], [(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)], [(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60)], [(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)], [(1,35,79,45),(2,24,80,54),(3,33,61,43),(4,22,62,52),(5,31,63,41),(6,40,64,50),(7,29,65,59),(8,38,66,48),(9,27,67,57),(10,36,68,46),(11,25,69,55),(12,34,70,44),(13,23,71,53),(14,32,72,42),(15,21,73,51),(16,30,74,60),(17,39,75,49),(18,28,76,58),(19,37,77,47),(20,26,78,56)]])
C23.D10 is a maximal subgroup of
C24.30D10 C24.31D10 C42.89D10 C42.93D10 C42.94D10 C42.98D10 C42.102D10 C42.104D10 C42.105D10 C42.106D10 C42.229D10 C42.113D10 C42.115D10 C42.118D10 C24.32D10 C24.35D10 C24.36D10 C4⋊C4.178D10 C10.342+ 1+4 C10.352+ 1+4 C10.362+ 1+4 C10.422+ 1+4 C10.432+ 1+4 C10.1152+ 1+4 C10.482+ 1+4 C10.152- 1+4 C10.202- 1+4 C10.212- 1+4 C10.222- 1+4 C10.232- 1+4 C10.582+ 1+4 C4⋊C4.197D10 C10.802- 1+4 C10.812- 1+4 C10.612+ 1+4 C10.622+ 1+4 C10.632+ 1+4 C10.642+ 1+4 C10.842- 1+4 C10.852- 1+4 C42.137D10 C42.139D10 C42.140D10 C42⋊20D10 C42⋊21D10 C42.234D10 C42.144D10 C42.159D10 C42.160D10 D5×C42⋊2C2 C42⋊24D10 C42.162D10 C42.165D10 D6⋊C4.D5 C60⋊5C4⋊C2 D6⋊Dic5.C2 C5⋊(C42⋊3S3) C23.13(S3×D5) C23.14(S3×D5) C23.8D30
C23.D10 is a maximal quotient of
C5⋊2(C42⋊5C4) C10.51(C4×D4) C2.(C4×D20) C10.52(C4×D4) (C2×Dic5).Q8 (C2×C20).28D4 (C2×C4).Dic10 (C22×C4).D10 C24.3D10 C24.4D10 C24.6D10 C24.8D10 C24.9D10 C23.14D20 D6⋊C4.D5 C60⋊5C4⋊C2 D6⋊Dic5.C2 C5⋊(C42⋊3S3) C23.13(S3×D5) C23.14(S3×D5) C23.8D30
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 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 | 4 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | C4○D20 | D4⋊2D5 |
| kernel | C23.D10 | C4×Dic5 | C10.D4 | C4⋊Dic5 | C23.D5 | C5×C22⋊C4 | C22⋊C4 | C10 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 6 | 4 | 2 | 8 | 4 |
Matrix representation of C23.D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 25 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 8 | 0 | 0 | 0 |
| 9 | 5 | 0 | 0 |
| 0 | 0 | 40 | 5 |
| 0 | 0 | 0 | 1 |
| 30 | 31 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [1,6,0,0,0,40,0,0,0,0,1,25,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[8,9,0,0,0,5,0,0,0,0,40,0,0,0,5,1],[30,4,0,0,31,11,0,0,0,0,32,0,0,0,0,32] >;
C23.D10 in GAP, Magma, Sage, TeX
C_2^3.D_{10} % in TeX
G:=Group("C2^3.D10"); // GroupNames label
G:=SmallGroup(160,100);
// by ID
G=gap.SmallGroup(160,100);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,506,188,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^10=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations