metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.21D10, (C2×C20)⋊11C4, (C2×C4)⋊4Dic5, C20.58(C2×C4), C4⋊Dic5⋊17C2, (C22×C4).7D5, C5⋊5(C42⋊C2), (C4×Dic5)⋊15C2, C2.4(C4○D20), (C2×C4).102D10, C4.15(C2×Dic5), C23.D5.5C2, C10.16(C4○D4), C10.37(C22×C4), (C2×C10).44C23, (C22×C20).10C2, (C2×C20).93C22, C22.5(C2×Dic5), C2.5(C22×Dic5), C22.22(C22×D5), (C22×C10).36C22, (C2×Dic5).38C22, (C2×C10).55(C2×C4), SmallGroup(160,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.21D10
G = < a,b,c,d,e | a2=b2=c2=1, d10=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >
Subgroups: 168 in 76 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C22×C20, C23.21D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C42⋊C2, C2×Dic5 [×6], C22×D5, C4○D20 [×2], C22×Dic5, C23.21D10
►On 80 points
(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)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)
(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 29 65 49)(2 38 66 58)(3 27 67 47)(4 36 68 56)(5 25 69 45)(6 34 70 54)(7 23 71 43)(8 32 72 52)(9 21 73 41)(10 30 74 50)(11 39 75 59)(12 28 76 48)(13 37 77 57)(14 26 78 46)(15 35 79 55)(16 24 80 44)(17 33 61 53)(18 22 62 42)(19 31 63 51)(20 40 64 60)
G:=sub<Sym(80)| (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), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50), (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), (1,29,65,49)(2,38,66,58)(3,27,67,47)(4,36,68,56)(5,25,69,45)(6,34,70,54)(7,23,71,43)(8,32,72,52)(9,21,73,41)(10,30,74,50)(11,39,75,59)(12,28,76,48)(13,37,77,57)(14,26,78,46)(15,35,79,55)(16,24,80,44)(17,33,61,53)(18,22,62,42)(19,31,63,51)(20,40,64,60)>;
G:=Group( (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), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50), (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), (1,29,65,49)(2,38,66,58)(3,27,67,47)(4,36,68,56)(5,25,69,45)(6,34,70,54)(7,23,71,43)(8,32,72,52)(9,21,73,41)(10,30,74,50)(11,39,75,59)(12,28,76,48)(13,37,77,57)(14,26,78,46)(15,35,79,55)(16,24,80,44)(17,33,61,53)(18,22,62,42)(19,31,63,51)(20,40,64,60) );
G=PermutationGroup([(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)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50)], [(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)], [(1,29,65,49),(2,38,66,58),(3,27,67,47),(4,36,68,56),(5,25,69,45),(6,34,70,54),(7,23,71,43),(8,32,72,52),(9,21,73,41),(10,30,74,50),(11,39,75,59),(12,28,76,48),(13,37,77,57),(14,26,78,46),(15,35,79,55),(16,24,80,44),(17,33,61,53),(18,22,62,42),(19,31,63,51),(20,40,64,60)])
C23.21D10 is a maximal subgroup of
C42⋊6Dic5 C42⋊1Dic5 C20.33C42 (C2×C40)⋊C4 C23.9D20 M4(2)⋊4Dic5 C40⋊8C4⋊C2 C23.10D20 C23.13D20 C20.47(C4⋊C4) C20.76(C4⋊C4) (C2×D20)⋊25C4 (C2×D4).D10 C10.(C4○D8) C20.42C42 C20.65(C4⋊C4) C23.22D20 C23.23D20 C20.51(C4⋊C4) C23.46D20 C23.47D20 C20.37C42 C23.48D20 C23.20D20 (D4×C10)⋊18C4 (Q8×C10)⋊16C4 C20.(C2×D4) (D4×C10)⋊21C4 (D4×C10)⋊22C4 C42.274D10 C4×C4○D20 C24.31D10 C10.12- 1+4 C10.82+ 1+4 C10.52- 1+4 C42.87D10 C42.88D10 C42.90D10 D5×C42⋊C2 C42⋊7D10 C42.102D10 C42.105D10 C42.106D10 C42.229D10 C42.117D10 C42.119D10 C10.362+ 1+4 C10.432+ 1+4 C10.452+ 1+4 C10.472+ 1+4 C10.152- 1+4 C10.212- 1+4 C10.232- 1+4 C10.242- 1+4 C10.802- 1+4 C10.1222+ 1+4 C24.72D10 C24.38D10 C24.41D10 C10.422- 1+4 C10.442- 1+4 C10.1052- 1+4 C4○D4×Dic5 C10.1062- 1+4 (C2×C20)⋊15D4 C10.1462+ 1+4 (S3×C20)⋊5C4 (S3×C20)⋊7C4 C23.26(S3×D5) C23.26D30
C23.21D10 is a maximal quotient of
C42.6Dic5 C42.7Dic5 C42⋊4Dic5 C4×C4⋊Dic5 C42⋊9Dic5 C42⋊5Dic5 C24.8D10 C4⋊C4⋊5Dic5 C42.187D10 C4×C23.D5 C24.63D10 C24.64D10 (S3×C20)⋊5C4 (S3×C20)⋊7C4 C23.26(S3×D5) C23.26D30
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | Dic5 | D10 | D10 | C4○D20 |
| kernel | C23.21D10 | C4×Dic5 | C4⋊Dic5 | C23.D5 | C22×C20 | C2×C20 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 2 | 4 | 8 | 4 | 2 | 16 |
Matrix representation of C23.21D10 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 20 | 40 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 30 | 33 |
| 9 | 0 | 0 |
| 0 | 4 | 16 |
| 0 | 22 | 37 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,20,0,0,40],[40,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,40],[1,0,0,0,36,30,0,0,33],[9,0,0,0,4,22,0,16,37] >;
C23.21D10 in GAP, Magma, Sage, TeX
C_2^3._{21}D_{10} % in TeX
G:=Group("C2^3.21D10"); // GroupNames label
G:=SmallGroup(160,147);
// by ID
G=gap.SmallGroup(160,147);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,362,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^10=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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