metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.23D10, (C22×C4)⋊3D5, (C22×C20)⋊2C2, C10.42(C2×D4), (C2×C10).37D4, (C2×C4).65D10, C23.D5⋊6C2, D10⋊C4⋊2C2, C10.D4⋊3C2, C10.18(C4○D4), C2.18(C4○D20), (C2×C10).47C23, (C2×C20).78C22, C5⋊4(C22.D4), C22.9(C5⋊D4), (C22×D5).9C22, C22.55(C22×D5), (C22×C10).39C22, (C2×Dic5).15C22, C2.6(C2×C5⋊D4), (C2×C5⋊D4).6C2, SmallGroup(160,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.23D10
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=bd9 >
Subgroups: 240 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C5, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22.D4, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C10.D4 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4, C22×C20, C23.23D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C5⋊D4 [×2], C22×D5, C4○D20 [×2], C2×C5⋊D4, C23.23D10
►On 80 points
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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 10 66 75)(2 74 67 9)(3 8 68 73)(4 72 69 7)(5 6 70 71)(11 20 76 65)(12 64 77 19)(13 18 78 63)(14 62 79 17)(15 16 80 61)(21 48 59 30)(22 29 60 47)(23 46 41 28)(24 27 42 45)(25 44 43 26)(31 58 49 40)(32 39 50 57)(33 56 51 38)(34 37 52 55)(35 54 53 36)
G:=sub<Sym(80)| (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70), (1,76)(2,77)(3,78)(4,79)(5,80)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,10,66,75)(2,74,67,9)(3,8,68,73)(4,72,69,7)(5,6,70,71)(11,20,76,65)(12,64,77,19)(13,18,78,63)(14,62,79,17)(15,16,80,61)(21,48,59,30)(22,29,60,47)(23,46,41,28)(24,27,42,45)(25,44,43,26)(31,58,49,40)(32,39,50,57)(33,56,51,38)(34,37,52,55)(35,54,53,36)>;
G:=Group( (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70), (1,76)(2,77)(3,78)(4,79)(5,80)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,10,66,75)(2,74,67,9)(3,8,68,73)(4,72,69,7)(5,6,70,71)(11,20,76,65)(12,64,77,19)(13,18,78,63)(14,62,79,17)(15,16,80,61)(21,48,59,30)(22,29,60,47)(23,46,41,28)(24,27,42,45)(25,44,43,26)(31,58,49,40)(32,39,50,57)(33,56,51,38)(34,37,52,55)(35,54,53,36) );
G=PermutationGroup([(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,10,66,75),(2,74,67,9),(3,8,68,73),(4,72,69,7),(5,6,70,71),(11,20,76,65),(12,64,77,19),(13,18,78,63),(14,62,79,17),(15,16,80,61),(21,48,59,30),(22,29,60,47),(23,46,41,28),(24,27,42,45),(25,44,43,26),(31,58,49,40),(32,39,50,57),(33,56,51,38),(34,37,52,55),(35,54,53,36)])
C23.23D10 is a maximal subgroup of
(C22×C4)⋊F5 C22⋊C4⋊D10 C42.277D10 C24.27D10 C24.31D10 C10.2- 1+4 C10.52- 1+4 C10.62- 1+4 C42⋊10D10 C42.96D10 C42.104D10 C42⋊16D10 C42.113D10 C42.114D10 C42⋊17D10 C42.115D10 C42.116D10 C42.118D10 C10.422+ 1+4 C10.442+ 1+4 C10.482+ 1+4 C10.742- 1+4 C10.202- 1+4 C10.222- 1+4 C10.582+ 1+4 C10.262- 1+4 C10.792- 1+4 C4⋊C4.197D10 D5×C22.D4 C10.1202+ 1+4 C4⋊C4⋊28D10 C10.852- 1+4 C24.72D10 C24⋊8D10 C10.442- 1+4 C10.1042- 1+4 C10.1452+ 1+4 D6⋊Dic5⋊C2 D10⋊C4⋊S3 (C2×C30).D4 C10.(C2×D12) C23.28D30
C23.23D10 is a maximal quotient of
C10.92(C4×D4) (C2×C42)⋊D5 C24.9D10 C24.14D10 (C2×C10).40D8 C4⋊C4.228D10 C4⋊C4.230D10 C4⋊C4.231D10 (C2×C20).287D4 (C2×C20).288D4 (C2×C20).289D4 (C2×C20).290D4 C4⋊C4.233D10 C4⋊C4.236D10 C24.62D10 C24.63D10 C24.65D10 D6⋊Dic5⋊C2 D10⋊C4⋊S3 (C2×C30).D4 C10.(C2×D12) C23.28D30
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C4○D20 |
| kernel | C23.23D10 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C22×C20 | C2×C10 | C22×C4 | C10 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 8 | 16 |
Matrix representation of C23.23D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 40 | 17 |
| 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 |
| 35 | 26 | 0 | 0 |
| 38 | 20 | 0 | 0 |
| 0 | 0 | 19 | 19 |
| 0 | 0 | 22 | 9 |
| 21 | 26 | 0 | 0 |
| 24 | 20 | 0 | 0 |
| 0 | 0 | 19 | 19 |
| 0 | 0 | 9 | 22 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,24,40,0,0,1,17],[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],[35,38,0,0,26,20,0,0,0,0,19,22,0,0,19,9],[21,24,0,0,26,20,0,0,0,0,19,9,0,0,19,22] >;
C23.23D10 in GAP, Magma, Sage, TeX
C_2^3._{23}D_{10} % in TeX
G:=Group("C2^3.23D10"); // GroupNames label
G:=SmallGroup(160,150);
// by ID
G=gap.SmallGroup(160,150);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,86,4613]);
// 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=b*d^9>;
// generators/relations