metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.72D10, (C2×C20)⋊38D4, (C23×C4)⋊6D5, C20⋊7D4⋊51C2, (C23×C20)⋊10C2, C20.425(C2×D4), (C2×D20)⋊51C22, C22⋊5(C4○D20), C24⋊2D5⋊15C2, C4⋊Dic5⋊65C22, C20.48D4⋊51C2, (C2×C10).289C24, (C2×C20).887C23, C5⋊7(C22.19C24), (C4×Dic5)⋊59C22, C10.135(C22×D4), (C22×C4).449D10, D10⋊C4⋊43C22, (C2×Dic10)⋊59C22, C10.D4⋊45C22, C22.304(C23×D5), C23.235(C22×D5), C23.21D10⋊13C2, C23.23D10⋊33C2, (C22×C10).418C23, (C22×C20).530C22, (C23×C10).111C22, (C2×Dic5).151C23, (C22×D5).127C23, C23.D5.130C22, (C4×C5⋊D4)⋊51C2, (C2×C4×D5)⋊54C22, (C2×C4○D20)⋊14C2, (C2×C4)⋊17(C5⋊D4), C10.64(C2×C4○D4), C2.72(C2×C4○D20), C4.145(C2×C5⋊D4), (C2×C10)⋊12(C4○D4), C2.8(C22×C5⋊D4), (C2×C10).575(C2×D4), C22.35(C2×C5⋊D4), (C2×C4).740(C22×D5), (C2×C5⋊D4).145C22, SmallGroup(320,1463)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1022 in 330 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D5 [×2], C10, C10 [×2], C10 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic5 [×6], C20 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], C2×C10 [×14], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×6], C2×C20 [×10], C22×D5 [×2], C22×C10, C22×C10 [×2], C22×C10 [×6], C22.19C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×6], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×6], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C20.48D4 [×2], C23.21D10, C4×C5⋊D4 [×4], C23.23D10 [×2], C20⋊7D4 [×2], C24⋊2D5 [×2], C2×C4○D20, C23×C20, C24.72D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C22.19C24, C4○D20 [×4], C2×C5⋊D4 [×6], C23×D5, C2×C4○D20 [×2], C22×C5⋊D4, C24.72D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, ac=ca, faf-1=ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
►On 80 points
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(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 72 11 62)(2 61 12 71)(3 70 13 80)(4 79 14 69)(5 68 15 78)(6 77 16 67)(7 66 17 76)(8 75 18 65)(9 64 19 74)(10 73 20 63)(21 55 31 45)(22 44 32 54)(23 53 33 43)(24 42 34 52)(25 51 35 41)(26 60 36 50)(27 49 37 59)(28 58 38 48)(29 47 39 57)(30 56 40 46)
G:=sub<Sym(80)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (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), (1,72,11,62)(2,61,12,71)(3,70,13,80)(4,79,14,69)(5,68,15,78)(6,77,16,67)(7,66,17,76)(8,75,18,65)(9,64,19,74)(10,73,20,63)(21,55,31,45)(22,44,32,54)(23,53,33,43)(24,42,34,52)(25,51,35,41)(26,60,36,50)(27,49,37,59)(28,58,38,48)(29,47,39,57)(30,56,40,46)>;
G:=Group( (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (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), (1,72,11,62)(2,61,12,71)(3,70,13,80)(4,79,14,69)(5,68,15,78)(6,77,16,67)(7,66,17,76)(8,75,18,65)(9,64,19,74)(10,73,20,63)(21,55,31,45)(22,44,32,54)(23,53,33,43)(24,42,34,52)(25,51,35,41)(26,60,36,50)(27,49,37,59)(28,58,38,48)(29,47,39,57)(30,56,40,46) );
G=PermutationGroup([(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,21),(19,22),(20,23),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,21),(19,22),(20,23),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70)], [(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)], [(1,72,11,62),(2,61,12,71),(3,70,13,80),(4,79,14,69),(5,68,15,78),(6,77,16,67),(7,66,17,76),(8,75,18,65),(9,64,19,74),(10,73,20,63),(21,55,31,45),(22,44,32,54),(23,53,33,43),(24,42,34,52),(25,51,35,41),(26,60,36,50),(27,49,37,59),(28,58,38,48),(29,47,39,57),(30,56,40,46)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,8,0,0,0,0,5],[0,40,0,0,40,0,0,0,0,0,0,8,0,0,5,0] >;
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C4○D20 |
| kernel | C24.72D10 | C20.48D4 | C23.21D10 | C4×C5⋊D4 | C23.23D10 | C20⋊7D4 | C24⋊2D5 | C2×C4○D20 | C23×C20 | C2×C20 | C23×C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 2 | 8 | 12 | 2 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_2^4._{72}D_{10} % in TeX
G:=Group("C2^4.72D10"); // GroupNames label
G:=SmallGroup(320,1463);
// by ID
G=gap.SmallGroup(320,1463);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,12550]);
// 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,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*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