metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.18D10, (C2×D4).5D5, (C2×C10).7D4, (C2×C4).18D10, C10.47(C2×D4), C23.D5⋊8C2, (D4×C10).10C2, C10.29(C4○D4), C10.D4⋊14C2, (C2×C20).61C22, (C2×C10).50C23, (C22×Dic5)⋊5C2, C22.4(C5⋊D4), C5⋊5(C22.D4), C2.15(D4⋊2D5), C22.57(C22×D5), (C22×C10).18C22, (C2×Dic5).17C22, C2.11(C2×C5⋊D4), SmallGroup(160,156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.18D10
G = < a,b,c,d,e | a2=b2=c2=d10=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 208 in 78 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C22.D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C10.D4, C23.D5, C23.D5, C22×Dic5, D4×C10, C23.18D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C5⋊D4, C22×D5, D4⋊2D5, C2×C5⋊D4, C23.18D10
►On 80 points
(1 21)(2 27)(3 23)(4 29)(5 25)(6 24)(7 30)(8 26)(9 22)(10 28)(11 37)(12 33)(13 39)(14 35)(15 31)(16 36)(17 32)(18 38)(19 34)(20 40)(41 71)(42 62)(43 73)(44 64)(45 75)(46 66)(47 77)(48 68)(49 79)(50 70)(51 80)(52 61)(53 72)(54 63)(55 74)(56 65)(57 76)(58 67)(59 78)(60 69)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 16)(7 17)(8 18)(9 19)(10 20)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 17)(12 18)(13 19)(14 20)(15 16)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(49 60)(50 51)(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 56 18 50)(2 60 19 44)(3 54 20 48)(4 58 16 42)(5 52 17 46)(6 47 15 53)(7 41 11 57)(8 45 12 51)(9 49 13 55)(10 43 14 59)(21 75 38 80)(22 69 39 64)(23 73 40 78)(24 67 31 62)(25 71 32 76)(26 65 33 70)(27 79 34 74)(28 63 35 68)(29 77 36 72)(30 61 37 66)
G:=sub<Sym(80)| (1,21)(2,27)(3,23)(4,29)(5,25)(6,24)(7,30)(8,26)(9,22)(10,28)(11,37)(12,33)(13,39)(14,35)(15,31)(16,36)(17,32)(18,38)(19,34)(20,40)(41,71)(42,62)(43,73)(44,64)(45,75)(46,66)(47,77)(48,68)(49,79)(50,70)(51,80)(52,61)(53,72)(54,63)(55,74)(56,65)(57,76)(58,67)(59,78)(60,69), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,8)(2,9)(3,10)(4,6)(5,7)(11,17)(12,18)(13,19)(14,20)(15,16)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(49,60)(50,51)(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), (1,56,18,50)(2,60,19,44)(3,54,20,48)(4,58,16,42)(5,52,17,46)(6,47,15,53)(7,41,11,57)(8,45,12,51)(9,49,13,55)(10,43,14,59)(21,75,38,80)(22,69,39,64)(23,73,40,78)(24,67,31,62)(25,71,32,76)(26,65,33,70)(27,79,34,74)(28,63,35,68)(29,77,36,72)(30,61,37,66)>;
G:=Group( (1,21)(2,27)(3,23)(4,29)(5,25)(6,24)(7,30)(8,26)(9,22)(10,28)(11,37)(12,33)(13,39)(14,35)(15,31)(16,36)(17,32)(18,38)(19,34)(20,40)(41,71)(42,62)(43,73)(44,64)(45,75)(46,66)(47,77)(48,68)(49,79)(50,70)(51,80)(52,61)(53,72)(54,63)(55,74)(56,65)(57,76)(58,67)(59,78)(60,69), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,8)(2,9)(3,10)(4,6)(5,7)(11,17)(12,18)(13,19)(14,20)(15,16)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(49,60)(50,51)(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), (1,56,18,50)(2,60,19,44)(3,54,20,48)(4,58,16,42)(5,52,17,46)(6,47,15,53)(7,41,11,57)(8,45,12,51)(9,49,13,55)(10,43,14,59)(21,75,38,80)(22,69,39,64)(23,73,40,78)(24,67,31,62)(25,71,32,76)(26,65,33,70)(27,79,34,74)(28,63,35,68)(29,77,36,72)(30,61,37,66) );
G=PermutationGroup([[(1,21),(2,27),(3,23),(4,29),(5,25),(6,24),(7,30),(8,26),(9,22),(10,28),(11,37),(12,33),(13,39),(14,35),(15,31),(16,36),(17,32),(18,38),(19,34),(20,40),(41,71),(42,62),(43,73),(44,64),(45,75),(46,66),(47,77),(48,68),(49,79),(50,70),(51,80),(52,61),(53,72),(54,63),(55,74),(56,65),(57,76),(58,67),(59,78),(60,69)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,16),(7,17),(8,18),(9,19),(10,20),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,31),(30,32),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,17),(12,18),(13,19),(14,20),(15,16),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59),(49,60),(50,51),(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)], [(1,56,18,50),(2,60,19,44),(3,54,20,48),(4,58,16,42),(5,52,17,46),(6,47,15,53),(7,41,11,57),(8,45,12,51),(9,49,13,55),(10,43,14,59),(21,75,38,80),(22,69,39,64),(23,73,40,78),(24,67,31,62),(25,71,32,76),(26,65,33,70),(27,79,34,74),(28,63,35,68),(29,77,36,72),(30,61,37,66)]])
C23.18D10 is a maximal subgroup of
(C2×C20).D4 C23.4D20 C23.5D20 2+ 1+4.2D5 C42.102D10 C42.105D10 C42⋊16D10 C42.118D10 C24.56D10 C24.32D10 C24⋊3D10 C24.35D10 C24.36D10 C4⋊C4.178D10 C10.342+ 1+4 C10.352+ 1+4 C10.362+ 1+4 C10.402+ 1+4 C10.732- 1+4 C10.422+ 1+4 C10.432+ 1+4 C10.442+ 1+4 C10.742- 1+4 C10.792- 1+4 C10.802- 1+4 C10.812- 1+4 D5×C22.D4 C10.632+ 1+4 C10.672+ 1+4 C42.137D10 C42.140D10 C42⋊21D10 C42.166D10 C42.168D10 C42⋊28D10 C24⋊8D10 C24.42D10 C10.1042- 1+4 C10.1052- 1+4 (C2×C20)⋊15D4 (C2×D12).D5 C30.(C2×D4) (C2×C10).D12 (S3×C10).D4 C23.22D30
C23.18D10 is a maximal quotient of
C23.42D20 C24.3D10 C24.46D10 C24.7D10 C24.9D10 C23.14D20 C10.97(C4×D4) (C2×C20).53D4 (C2×C20).55D4 (C2×C10).D8 C4⋊D4.D5 (C2×D4).D10 C22⋊Q8.D5 (C2×C10).Q16 C10.(C4○D8) C24.18D10 C24.20D10 (C2×D12).D5 C30.(C2×D4) (C2×C10).D12 (S3×C10).D4 C23.22D30
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D4⋊2D5 |
| kernel | C23.18D10 | C10.D4 | C23.D5 | C22×Dic5 | D4×C10 | C2×C10 | C2×D4 | C10 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 4 |
Matrix representation of C23.18D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 22 |
| 0 | 0 | 0 | 1 |
| 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 |
| 16 | 0 | 0 | 0 |
| 19 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 40 |
| 18 | 39 | 0 | 0 |
| 19 | 23 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 12 | 32 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,22,1],[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],[16,19,0,0,0,23,0,0,0,0,1,15,0,0,0,40],[18,19,0,0,39,23,0,0,0,0,9,12,0,0,0,32] >;
C23.18D10 in GAP, Magma, Sage, TeX
C_2^3._{18}D_{10} % in TeX
G:=Group("C2^3.18D10"); // GroupNames label
G:=SmallGroup(160,156);
// by ID
G=gap.SmallGroup(160,156);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,188,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^10=1,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-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=b*d^-1>;
// generators/relations