direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.1D10, C23.15D20, C24.11D10, (C23×D5)⋊4C4, C10⋊3(C23⋊C4), C22⋊C4⋊36D10, C23.17(C4×D5), (C22×Dic5)⋊5C4, (C22×C10).66D4, C22.14(C2×D20), C23.74(C5⋊D4), C23.D5⋊42C22, C23.72(C22×D5), (C23×C10).37C22, (C22×C10).111C23, C22.44(D10⋊C4), C5⋊5(C2×C23⋊C4), (C2×C5⋊D4)⋊8C4, (C2×C22⋊C4)⋊1D5, C22.18(C2×C4×D5), (C10×C22⋊C4)⋊1C2, (C2×Dic5)⋊2(C2×C4), (C22×D5)⋊2(C2×C4), (C2×C23.D5)⋊1C2, (C2×C10).433(C2×D4), C2.8(C2×D10⋊C4), C10.76(C2×C22⋊C4), (C22×C5⋊D4).1C2, C22.26(C2×C5⋊D4), (C5×C22⋊C4)⋊44C22, (C2×C5⋊D4).91C22, (C22×C10).120(C2×C4), (C2×C10).113(C22×C4), (C2×C10).120(C22⋊C4), SmallGroup(320,581)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.1D10
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 926 in 210 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C2×C23⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C23.1D10, C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C2×C23.1D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C23⋊C4, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C2×C23⋊C4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C23.1D10, C2×D10⋊C4, C2×C23.1D10
►On 80 points
Generators in S80
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(1 11)(2 53)(3 13)(4 55)(5 15)(6 57)(7 17)(8 59)(9 19)(10 41)(12 43)(14 45)(16 47)(18 49)(20 51)(21 31)(22 77)(23 33)(24 79)(25 35)(26 61)(27 37)(28 63)(29 39)(30 65)(32 67)(34 69)(36 71)(38 73)(40 75)(42 52)(44 54)(46 56)(48 58)(50 60)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 41)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 61)(37 62)(38 63)(39 64)(40 65)
(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 51 11 20)(2 60 53 50)(3 8 13 59)(4 17 55 7)(5 47 15 16)(6 56 57 46)(9 43 19 12)(10 52 41 42)(14 48 45 58)(18 44 49 54)(21 38 31 73)(22 27 77 37)(23 61 33 26)(24 70 79 80)(25 34 35 69)(28 66 63 76)(29 30 39 65)(32 62 67 72)(36 78 71 68)(40 74 75 64)
G:=sub<Sym(80)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,11)(2,53)(3,13)(4,55)(5,15)(6,57)(7,17)(8,59)(9,19)(10,41)(12,43)(14,45)(16,47)(18,49)(20,51)(21,31)(22,77)(23,33)(24,79)(25,35)(26,61)(27,37)(28,63)(29,39)(30,65)(32,67)(34,69)(36,71)(38,73)(40,75)(42,52)(44,54)(46,56)(48,58)(50,60)(62,72)(64,74)(66,76)(68,78)(70,80), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,61)(37,62)(38,63)(39,64)(40,65), (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), (1,51,11,20)(2,60,53,50)(3,8,13,59)(4,17,55,7)(5,47,15,16)(6,56,57,46)(9,43,19,12)(10,52,41,42)(14,48,45,58)(18,44,49,54)(21,38,31,73)(22,27,77,37)(23,61,33,26)(24,70,79,80)(25,34,35,69)(28,66,63,76)(29,30,39,65)(32,62,67,72)(36,78,71,68)(40,74,75,64)>;
G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,11)(2,53)(3,13)(4,55)(5,15)(6,57)(7,17)(8,59)(9,19)(10,41)(12,43)(14,45)(16,47)(18,49)(20,51)(21,31)(22,77)(23,33)(24,79)(25,35)(26,61)(27,37)(28,63)(29,39)(30,65)(32,67)(34,69)(36,71)(38,73)(40,75)(42,52)(44,54)(46,56)(48,58)(50,60)(62,72)(64,74)(66,76)(68,78)(70,80), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,61)(37,62)(38,63)(39,64)(40,65), (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), (1,51,11,20)(2,60,53,50)(3,8,13,59)(4,17,55,7)(5,47,15,16)(6,56,57,46)(9,43,19,12)(10,52,41,42)(14,48,45,58)(18,44,49,54)(21,38,31,73)(22,27,77,37)(23,61,33,26)(24,70,79,80)(25,34,35,69)(28,66,63,76)(29,30,39,65)(32,62,67,72)(36,78,71,68)(40,74,75,64) );
G=PermutationGroup([[(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(41,79),(42,80),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)], [(1,11),(2,53),(3,13),(4,55),(5,15),(6,57),(7,17),(8,59),(9,19),(10,41),(12,43),(14,45),(16,47),(18,49),(20,51),(21,31),(22,77),(23,33),(24,79),(25,35),(26,61),(27,37),(28,63),(29,39),(30,65),(32,67),(34,69),(36,71),(38,73),(40,75),(42,52),(44,54),(46,56),(48,58),(50,60),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,41),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,61),(37,62),(38,63),(39,64),(40,65)], [(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)], [(1,51,11,20),(2,60,53,50),(3,8,13,59),(4,17,55,7),(5,47,15,16),(6,56,57,46),(9,43,19,12),(10,52,41,42),(14,48,45,58),(18,44,49,54),(21,38,31,73),(22,27,77,37),(23,61,33,26),(24,70,79,80),(25,34,35,69),(28,66,63,76),(29,30,39,65),(32,62,67,72),(36,78,71,68),(40,74,75,64)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | C23.1D10 |
| kernel | C2×C23.1D10 | C23.1D10 | C2×C23.D5 | C10×C22⋊C4 | C22×C5⋊D4 | C22×Dic5 | C2×C5⋊D4 | C23×D5 | C22×C10 | C2×C22⋊C4 | C22⋊C4 | C24 | C23 | C23 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 8 | 8 | 8 | 2 | 8 |
Matrix representation of C2×C23.1D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 39 | 9 | 0 | 40 |
| 0 | 0 | 2 | 32 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 22 | 3 | 0 | 0 | 0 | 0 |
| 38 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 0 | 0 | 0 | 40 | 32 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 3 | 19 | 0 | 0 | 0 | 0 |
| 19 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 32 | 0 |
| 0 | 0 | 2 | 0 | 40 | 1 |
| 0 | 0 | 0 | 40 | 32 | 0 |
| 0 | 0 | 23 | 0 | 9 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,2,0,0,9,1,9,32,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[22,38,0,0,0,0,3,22,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,32,40,32,9,0,0,0,40,0,0],[3,19,0,0,0,0,19,38,0,0,0,0,0,0,9,2,0,23,0,0,0,0,40,0,0,0,32,40,32,9,0,0,0,1,0,0] >;
C2×C23.1D10 in GAP, Magma, Sage, TeX
C_2\times C_2^3._1D_{10} % in TeX
G:=Group("C2xC2^3.1D10"); // GroupNames label
G:=SmallGroup(320,581);
// by ID
G=gap.SmallGroup(320,581);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,438,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations