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