metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.33D10, C10.272+ 1+4, C5⋊D4⋊5D4, (C2×D4)⋊19D10, C5⋊4(D4⋊5D4), C22≀C2⋊4D5, C22⋊C4⋊6D10, C23⋊D10⋊5C2, C20⋊2D4⋊13C2, (D4×Dic5)⋊12C2, D10.39(C2×D4), (D4×C10)⋊8C22, C22.11(D4×D5), Dic5⋊4D4⋊3C2, Dic5⋊D4⋊3C2, (C2×C20).29C23, C4⋊Dic5⋊26C22, Dic5.43(C2×D4), C10.57(C22×D4), C22⋊4(D4⋊2D5), C23.7(C22×D5), (C2×C10).135C24, (C4×Dic5)⋊15C22, (C22×C10).9C23, D10.12D4⋊13C2, C23.D5⋊50C22, C2.29(D4⋊6D10), D10⋊C4⋊12C22, Dic5.5D4⋊12C2, (C2×Dic10)⋊20C22, C10.D4⋊10C22, C22.D20⋊10C2, (C23×C10).68C22, (C22×D5).54C23, (C23×D5).43C22, C22.156(C23×D5), Dic5.14D4⋊13C2, (C2×Dic5).232C23, (C22×Dic5)⋊14C22, C2.30(C2×D4×D5), (C2×C4×D5)⋊8C22, (D5×C22⋊C4)⋊3C2, (C5×C22≀C2)⋊6C2, (C2×D4⋊2D5)⋊6C2, C10.77(C2×C4○D4), (C2×C10).54(C2×D4), (C22×C5⋊D4)⋊9C2, (C2×C5⋊D4)⋊8C22, (C2×C10)⋊10(C4○D4), C2.28(C2×D4⋊2D5), (C5×C22⋊C4)⋊6C22, (C2×C23.D5)⋊20C2, (C2×C4).29(C22×D5), SmallGroup(320,1263)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.33D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 1286 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, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, 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, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23×D5, C23×C10, Dic5.14D4, D5×C22⋊C4, Dic5⋊4D4, D10.12D4, Dic5.5D4, C22.D20, D4×Dic5, C23⋊D10, C20⋊2D4, Dic5⋊D4, C2×C23.D5, C5×C22≀C2, C2×D4⋊2D5, C22×C5⋊D4, C24.33D10
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, D4×D5, D4⋊2D5, C23×D5, C2×D4×D5, C2×D4⋊2D5, D4⋊6D10, C24.33D10
►On 80 points
(1 11)(2 73)(3 13)(4 75)(5 15)(6 77)(7 17)(8 79)(9 19)(10 61)(12 63)(14 65)(16 67)(18 69)(20 71)(21 31)(22 46)(23 33)(24 48)(25 35)(26 50)(27 37)(28 52)(29 39)(30 54)(32 56)(34 58)(36 60)(38 42)(40 44)(41 51)(43 53)(45 55)(47 57)(49 59)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 27)(2 38)(3 29)(4 40)(5 31)(6 22)(7 33)(8 24)(9 35)(10 26)(11 37)(12 28)(13 39)(14 30)(15 21)(16 32)(17 23)(18 34)(19 25)(20 36)(41 62)(42 73)(43 64)(44 75)(45 66)(46 77)(47 68)(48 79)(49 70)(50 61)(51 72)(52 63)(53 74)(54 65)(55 76)(56 67)(57 78)(58 69)(59 80)(60 71)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 61)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)
(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 20 11 10)(2 9 12 19)(3 18 13 8)(4 7 14 17)(5 16 15 6)(21 56 31 46)(22 45 32 55)(23 54 33 44)(24 43 34 53)(25 52 35 42)(26 41 36 51)(27 50 37 60)(28 59 38 49)(29 48 39 58)(30 57 40 47)(61 72 71 62)(63 70 73 80)(64 79 74 69)(65 68 75 78)(66 77 76 67)
G:=sub<Sym(80)| (1,11)(2,73)(3,13)(4,75)(5,15)(6,77)(7,17)(8,79)(9,19)(10,61)(12,63)(14,65)(16,67)(18,69)(20,71)(21,31)(22,46)(23,33)(24,48)(25,35)(26,50)(27,37)(28,52)(29,39)(30,54)(32,56)(34,58)(36,60)(38,42)(40,44)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,27)(2,38)(3,29)(4,40)(5,31)(6,22)(7,33)(8,24)(9,35)(10,26)(11,37)(12,28)(13,39)(14,30)(15,21)(16,32)(17,23)(18,34)(19,25)(20,36)(41,62)(42,73)(43,64)(44,75)(45,66)(46,77)(47,68)(48,79)(49,70)(50,61)(51,72)(52,63)(53,74)(54,65)(55,76)(56,67)(57,78)(58,69)(59,80)(60,71), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,61)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,56,31,46)(22,45,32,55)(23,54,33,44)(24,43,34,53)(25,52,35,42)(26,41,36,51)(27,50,37,60)(28,59,38,49)(29,48,39,58)(30,57,40,47)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)>;
G:=Group( (1,11)(2,73)(3,13)(4,75)(5,15)(6,77)(7,17)(8,79)(9,19)(10,61)(12,63)(14,65)(16,67)(18,69)(20,71)(21,31)(22,46)(23,33)(24,48)(25,35)(26,50)(27,37)(28,52)(29,39)(30,54)(32,56)(34,58)(36,60)(38,42)(40,44)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,27)(2,38)(3,29)(4,40)(5,31)(6,22)(7,33)(8,24)(9,35)(10,26)(11,37)(12,28)(13,39)(14,30)(15,21)(16,32)(17,23)(18,34)(19,25)(20,36)(41,62)(42,73)(43,64)(44,75)(45,66)(46,77)(47,68)(48,79)(49,70)(50,61)(51,72)(52,63)(53,74)(54,65)(55,76)(56,67)(57,78)(58,69)(59,80)(60,71), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,61)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,56,31,46)(22,45,32,55)(23,54,33,44)(24,43,34,53)(25,52,35,42)(26,41,36,51)(27,50,37,60)(28,59,38,49)(29,48,39,58)(30,57,40,47)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67) );
G=PermutationGroup([[(1,11),(2,73),(3,13),(4,75),(5,15),(6,77),(7,17),(8,79),(9,19),(10,61),(12,63),(14,65),(16,67),(18,69),(20,71),(21,31),(22,46),(23,33),(24,48),(25,35),(26,50),(27,37),(28,52),(29,39),(30,54),(32,56),(34,58),(36,60),(38,42),(40,44),(41,51),(43,53),(45,55),(47,57),(49,59),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,27),(2,38),(3,29),(4,40),(5,31),(6,22),(7,33),(8,24),(9,35),(10,26),(11,37),(12,28),(13,39),(14,30),(15,21),(16,32),(17,23),(18,34),(19,25),(20,36),(41,62),(42,73),(43,64),(44,75),(45,66),(46,77),(47,68),(48,79),(49,70),(50,61),(51,72),(52,63),(53,74),(54,65),(55,76),(56,67),(57,78),(58,69),(59,80),(60,71)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,61),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54)], [(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)], [(1,20,11,10),(2,9,12,19),(3,18,13,8),(4,7,14,17),(5,16,15,6),(21,56,31,46),(22,45,32,55),(23,54,33,44),(24,43,34,53),(25,52,35,42),(26,41,36,51),(27,50,37,60),(28,59,38,49),(29,48,39,58),(30,57,40,47),(61,72,71,62),(63,70,73,80),(64,79,74,69),(65,68,75,78),(66,77,76,67)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | ··· | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2+ 1+4 | D4×D5 | D4⋊2D5 | D4⋊6D10 |
| kernel | C24.33D10 | Dic5.14D4 | D5×C22⋊C4 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C22.D20 | D4×Dic5 | C23⋊D10 | C20⋊2D4 | Dic5⋊D4 | C2×C23.D5 | C5×C22≀C2 | C2×D4⋊2D5 | C22×C5⋊D4 | C5⋊D4 | C22≀C2 | C2×C10 | C22⋊C4 | C2×D4 | C24 | C10 | C22 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C24.33D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 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 | 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 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 | 9 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,32,0,0,0,0,0,9],[34,1,0,0,0,0,34,7,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
C24.33D10 in GAP, Magma, Sage, TeX
C_2^4._{33}D_{10} % in TeX
G:=Group("C2^4.33D10"); // GroupNames label
G:=SmallGroup(320,1263);
// by ID
G=gap.SmallGroup(320,1263);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*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