metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.27D10, C10.32+ 1+4, C5⋊D4⋊8D4, C5⋊1(D4⋊5D4), (C22×C4)⋊8D10, C22⋊D20⋊2C2, D10⋊D4⋊1C2, C20⋊7D4⋊17C2, C22⋊C4⋊40D10, D10.35(C2×D4), (C2×D20)⋊2C22, C24⋊2D5⋊2C2, C4⋊Dic5⋊4C22, C22.18(D4×D5), C22⋊4(C4○D20), (C2×C10).34C24, Dic5.38(C2×D4), C10.37(C22×D4), Dic5⋊4D4⋊40C2, D10.12D4⋊1C2, C23.D5⋊8C22, C2.7(D4⋊6D10), (C2×C20).128C23, (C22×C20)⋊14C22, Dic5.5D4⋊1C2, (C4×Dic5)⋊47C22, D10⋊C4⋊46C22, C22.73(C23×D5), Dic5.14D4⋊2C2, (C2×Dic10)⋊48C22, C23.23D10⋊9C2, C10.D4⋊49C22, (C23×C10).60C22, (C23×D5).31C22, C23.221(C22×D5), (C22×C10).387C23, (C2×Dic5).190C23, (C22×D5).162C23, (C22×Dic5).78C22, C2.11(C2×D4×D5), (C4×C5⋊D4)⋊1C2, (C2×C4○D20)⋊3C2, (C2×C4×D5)⋊40C22, (C2×C10)⋊8(C4○D4), (C2×C22⋊C4)⋊13D5, (D5×C22⋊C4)⋊24C2, C2.16(C2×C4○D20), C10.14(C2×C4○D4), (C22×C5⋊D4)⋊5C2, (C2×C5⋊D4)⋊1C22, (C10×C22⋊C4)⋊18C2, (C2×C10).383(C2×D4), (C5×C22⋊C4)⋊53C22, (C2×C4).259(C22×D5), SmallGroup(320,1162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.27D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 1334 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, D4⋊5D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, Dic5.14D4, D5×C22⋊C4, Dic5⋊4D4, C22⋊D20, D10.12D4, D10⋊D4, Dic5.5D4, C4×C5⋊D4, C23.23D10, C20⋊7D4, C24⋊2D5, C10×C22⋊C4, C2×C4○D20, C22×C5⋊D4, C24.27D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊5D4, C4○D20, D4×D5, C23×D5, C2×C4○D20, C2×D4×D5, D4⋊6D10, C24.27D10
►On 80 points
(1 11)(2 79)(3 13)(4 61)(5 15)(6 63)(7 17)(8 65)(9 19)(10 67)(12 69)(14 71)(16 73)(18 75)(20 77)(21 31)(22 57)(23 33)(24 59)(25 35)(26 41)(27 37)(28 43)(29 39)(30 45)(32 47)(34 49)(36 51)(38 53)(40 55)(42 52)(44 54)(46 56)(48 58)(50 60)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 41)(19 42)(20 43)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)
(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 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 41)(37 42)(38 43)(39 44)(40 45)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 74 71 64)(62 63 72 73)(65 70 75 80)(66 79 76 69)(67 68 77 78)
G:=sub<Sym(80)| (1,11)(2,79)(3,13)(4,61)(5,15)(6,63)(7,17)(8,65)(9,19)(10,67)(12,69)(14,71)(16,73)(18,75)(20,77)(21,31)(22,57)(23,33)(24,59)(25,35)(26,41)(27,37)(28,43)(29,39)(30,45)(32,47)(34,49)(36,51)(38,53)(40,55)(42,52)(44,54)(46,56)(48,58)(50,60)(62,72)(64,74)(66,76)(68,78)(70,80), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69), (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), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(40,45), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,74,71,64)(62,63,72,73)(65,70,75,80)(66,79,76,69)(67,68,77,78)>;
G:=Group( (1,11)(2,79)(3,13)(4,61)(5,15)(6,63)(7,17)(8,65)(9,19)(10,67)(12,69)(14,71)(16,73)(18,75)(20,77)(21,31)(22,57)(23,33)(24,59)(25,35)(26,41)(27,37)(28,43)(29,39)(30,45)(32,47)(34,49)(36,51)(38,53)(40,55)(42,52)(44,54)(46,56)(48,58)(50,60)(62,72)(64,74)(66,76)(68,78)(70,80), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69), (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), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(40,45), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,74,71,64)(62,63,72,73)(65,70,75,80)(66,79,76,69)(67,68,77,78) );
G=PermutationGroup([[(1,11),(2,79),(3,13),(4,61),(5,15),(6,63),(7,17),(8,65),(9,19),(10,67),(12,69),(14,71),(16,73),(18,75),(20,77),(21,31),(22,57),(23,33),(24,59),(25,35),(26,41),(27,37),(28,43),(29,39),(30,45),(32,47),(34,49),(36,51),(38,53),(40,55),(42,52),(44,54),(46,56),(48,58),(50,60),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,41),(19,42),(20,43),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69)], [(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)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,41),(37,42),(38,43),(39,44),(40,45)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,36,31,26),(22,25,32,35),(23,34,33,24),(27,30,37,40),(28,39,38,29),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(61,74,71,64),(62,63,72,73),(65,70,75,80),(66,79,76,69),(67,68,77,78)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4×D5 | D4⋊6D10 |
| kernel | C24.27D10 | Dic5.14D4 | D5×C22⋊C4 | Dic5⋊4D4 | C22⋊D20 | D10.12D4 | D10⋊D4 | Dic5.5D4 | C4×C5⋊D4 | C23.23D10 | C20⋊7D4 | C24⋊2D5 | C10×C22⋊C4 | C2×C4○D20 | C22×C5⋊D4 | C5⋊D4 | C2×C22⋊C4 | C2×C10 | C22⋊C4 | C22×C4 | C24 | C22 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 8 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C24.27D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 23 | 6 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 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 |
| 13 | 13 | 0 | 0 |
| 28 | 9 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 40 |
| 13 | 13 | 0 | 0 |
| 9 | 28 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,1,0,0,0,40],[23,35,0,0,6,18,0,0,0,0,40,0,0,0,0,40],[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],[13,28,0,0,13,9,0,0,0,0,1,0,0,0,39,40],[13,9,0,0,13,28,0,0,0,0,1,0,0,0,39,40] >;
C24.27D10 in GAP, Magma, Sage, TeX
C_2^4._{27}D_{10} % in TeX
G:=Group("C2^4.27D10"); // GroupNames label
G:=SmallGroup(320,1162);
// by ID
G=gap.SmallGroup(320,1162);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations