metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.24D10, C10.22+ (1+4), C23⋊3(C4×D5), (C22×C4)⋊3D10, C22⋊C4⋊51D10, C10.30(C23×C4), (C2×C10).30C24, Dic5⋊4D4⋊39C2, C2.1(D4⋊6D10), (C2×C20).571C23, (C22×C20)⋊34C22, C5⋊2(C22.11C24), (C4×Dic5)⋊46C22, D10.11(C22×C4), C23.D5⋊67C22, D10⋊C4⋊57C22, C22.19(C23×D5), C10.D4⋊58C22, (C23×C10).56C22, Dic5.11(C22×C4), (C22×Dic5)⋊5C22, (C23×D5).30C22, C23.220(C22×D5), C23.11D10⋊24C2, (C22×C10).122C23, (C2×Dic5).187C23, (C22×D5).160C23, C5⋊D4⋊13(C2×C4), (C4×C5⋊D4)⋊33C2, (C2×C5⋊D4)⋊14C4, (C2×C22⋊C4)⋊7D5, (C2×C4×D5)⋊39C22, C22.24(C2×C4×D5), C2.11(D5×C22×C4), (D5×C22⋊C4)⋊23C2, (C22×D5)⋊8(C2×C4), (C10×C22⋊C4)⋊26C2, (C22×C10)⋊17(C2×C4), (C2×Dic5)⋊12(C2×C4), (C2×C23.D5)⋊15C2, (C22×C5⋊D4).9C2, (C5×C22⋊C4)⋊61C22, (C2×C4).257(C22×D5), (C2×C5⋊D4).96C22, (C2×C10).119(C22×C4), SmallGroup(320,1158)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1166 in 338 conjugacy classes, 151 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×12], C22, C22 [×6], C22 [×22], C5, C2×C4 [×4], C2×C4 [×18], D4 [×16], C23 [×3], C23 [×4], C23 [×10], D5 [×4], C10, C10 [×2], C10 [×6], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×7], C2×D4 [×12], C24, C24, Dic5 [×4], Dic5 [×4], C20 [×4], D10 [×4], D10 [×8], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×3], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C4×D5 [×4], C2×Dic5 [×10], C2×Dic5 [×2], C5⋊D4 [×16], C2×C20 [×4], C2×C20 [×2], C22×D5 [×6], C22×D5 [×2], C22×C10 [×3], C22×C10 [×4], C22×C10 [×2], C22.11C24, C4×Dic5 [×4], C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×4], C5×C22⋊C4 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×12], C22×C20 [×2], C23×D5, C23×C10, C23.11D10 [×2], D5×C22⋊C4 [×2], Dic5⋊4D4 [×4], C4×C5⋊D4 [×4], C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C24.24D10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ (1+4) [×2], C4×D5 [×4], C22×D5 [×7], C22.11C24, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D4⋊6D10 [×2], C24.24D10
Generators and relations
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, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
►On 80 points
(2 21)(4 23)(6 25)(8 27)(10 29)(12 31)(14 33)(16 35)(18 37)(20 39)(41 75)(43 77)(45 79)(47 61)(49 63)(51 65)(53 67)(55 69)(57 71)(59 73)
(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 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)
(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 40)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)
(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 38 31 28)(22 27 32 37)(23 36 33 26)(24 25 34 35)(29 30 39 40)(41 80 51 70)(42 69 52 79)(43 78 53 68)(44 67 54 77)(45 76 55 66)(46 65 56 75)(47 74 57 64)(48 63 58 73)(49 72 59 62)(50 61 60 71)
G:=sub<Sym(80)| (2,21)(4,23)(6,25)(8,27)(10,29)(12,31)(14,33)(16,35)(18,37)(20,39)(41,75)(43,77)(45,79)(47,61)(49,63)(51,65)(53,67)(55,69)(57,71)(59,73), (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)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68), (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), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74), (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)(21,38,31,28)(22,27,32,37)(23,36,33,26)(24,25,34,35)(29,30,39,40)(41,80,51,70)(42,69,52,79)(43,78,53,68)(44,67,54,77)(45,76,55,66)(46,65,56,75)(47,74,57,64)(48,63,58,73)(49,72,59,62)(50,61,60,71)>;
G:=Group( (2,21)(4,23)(6,25)(8,27)(10,29)(12,31)(14,33)(16,35)(18,37)(20,39)(41,75)(43,77)(45,79)(47,61)(49,63)(51,65)(53,67)(55,69)(57,71)(59,73), (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)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68), (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), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74), (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)(21,38,31,28)(22,27,32,37)(23,36,33,26)(24,25,34,35)(29,30,39,40)(41,80,51,70)(42,69,52,79)(43,78,53,68)(44,67,54,77)(45,76,55,66)(46,65,56,75)(47,74,57,64)(48,63,58,73)(49,72,59,62)(50,61,60,71) );
G=PermutationGroup([(2,21),(4,23),(6,25),(8,27),(10,29),(12,31),(14,33),(16,35),(18,37),(20,39),(41,75),(43,77),(45,79),(47,61),(49,63),(51,65),(53,67),(55,69),(57,71),(59,73)], [(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),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68)], [(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)], [(1,40),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(57,71),(58,72),(59,73),(60,74)], [(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),(21,38,31,28),(22,27,32,37),(23,36,33,26),(24,25,34,35),(29,30,39,40),(41,80,51,70),(42,69,52,79),(43,78,53,68),(44,67,54,77),(45,76,55,66),(46,65,56,75),(47,74,57,64),(48,63,58,73),(49,72,59,62),(50,61,60,71)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 2 | 40 | 0 |
| 0 | 0 | 14 | 2 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 40 | 0 | 0 |
| 0 | 0 | 36 | 23 | 0 | 0 |
| 0 | 0 | 9 | 0 | 17 | 40 |
| 0 | 0 | 28 | 32 | 1 | 24 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 22 | 0 | 0 | 0 | 0 |
| 19 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 22 | 27 | 33 |
| 0 | 0 | 9 | 13 | 35 | 34 |
| 0 | 0 | 8 | 6 | 19 | 19 |
| 0 | 0 | 8 | 13 | 19 | 19 |
| 19 | 19 | 0 | 0 | 0 | 0 |
| 9 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 22 | 33 | 27 |
| 0 | 0 | 9 | 13 | 34 | 35 |
| 0 | 0 | 8 | 6 | 19 | 19 |
| 0 | 0 | 2 | 13 | 19 | 19 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,14,14,0,0,0,1,2,2,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,36,9,28,0,0,40,23,0,32,0,0,0,0,17,1,0,0,0,0,40,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[22,19,0,0,0,0,22,32,0,0,0,0,0,0,31,9,8,8,0,0,22,13,6,13,0,0,27,35,19,19,0,0,33,34,19,19],[19,9,0,0,0,0,19,22,0,0,0,0,0,0,31,9,8,2,0,0,22,13,6,13,0,0,33,34,19,19,0,0,27,35,19,19] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4T | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | C4×D5 | 2+ (1+4) | D4⋊6D10 |
| kernel | C24.24D10 | C23.11D10 | D5×C22⋊C4 | Dic5⋊4D4 | C4×C5⋊D4 | C2×C23.D5 | C10×C22⋊C4 | C22×C5⋊D4 | C2×C5⋊D4 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C10 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 16 | 2 | 8 | 4 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2^4._{24}D_{10} % in TeX
G:=Group("C2^4.24D10"); // GroupNames label
G:=SmallGroup(320,1158);
// by ID
G=gap.SmallGroup(320,1158);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,570,80,12550]);
// 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,b*c=c*b,f*b*f^-1=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