metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4⋊3F5, C4○D20⋊3C4, D4.9(C2×F5), Q8.9(C2×F5), D4⋊2D5⋊9C4, D20⋊C4⋊5C2, Q8⋊2D5⋊9C4, Q8⋊F5⋊5C2, D20.9(C2×C4), (C4×D5).56D4, D5.5(C4○D8), C5⋊(C23.24D4), C4⋊F5.10C22, C4.23(C22×F5), D10.101(C2×D4), C20.23(C22×C4), D5⋊C8.16C22, Dic10.9(C2×C4), Dic5.11(C2×D4), (C22×D5).73D4, (D4×D5).15C22, (C4×D5).45C23, C4.47(C22⋊F5), (Q8×D5).13C22, C20.47(C22⋊C4), (C2×Dic5).125D4, C22.6(C22⋊F5), D10.18(C22⋊C4), D10.C23⋊6C2, Dic5.50(C22⋊C4), (C2×D5⋊C8)⋊4C2, (C5×C4○D4)⋊3C4, (D5×C4○D4).7C2, (C5×D4).9(C2×C4), (C2×C4).90(C2×F5), (C5×Q8).9(C2×C4), (C2×C20).68(C2×C4), (C4×D5).29(C2×C4), C2.36(C2×C22⋊F5), C10.35(C2×C22⋊C4), (C2×C4×D5).212C22, (C2×C10).6(C22⋊C4), SmallGroup(320,1132)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 682 in 158 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×6], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×12], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], Dic5 [×2], Dic5, C20 [×2], C20, F5 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10, D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C5⋊C8 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5 [×2], C22×D5, C22×D5, C23.24D4, D5⋊C8 [×2], D5⋊C8, C4×F5, C4⋊F5 [×2], C2×C5⋊C8, C22⋊F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, D20⋊C4 [×2], Q8⋊F5 [×2], C2×D5⋊C8, D10.C23, D5×C4○D4, C4○D20⋊C4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C4○D8 [×2], C2×F5 [×3], C23.24D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, C4○D20⋊C4
Generators and relations
G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, ad=da, cbc=a2b9, dbd-1=b7, dcd-1=a2bc >
►On 80 points
(1 58 11 48)(2 59 12 49)(3 60 13 50)(4 41 14 51)(5 42 15 52)(6 43 16 53)(7 44 17 54)(8 45 18 55)(9 46 19 56)(10 47 20 57)(21 62 31 72)(22 63 32 73)(23 64 33 74)(24 65 34 75)(25 66 35 76)(26 67 36 77)(27 68 37 78)(28 69 38 79)(29 70 39 80)(30 71 40 61)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 40)(19 39)(20 38)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 80)(57 79)(58 78)(59 77)(60 76)
(1 6)(2 9 10 13)(3 12 19 20)(4 15 8 7)(5 18 17 14)(11 16)(21 37 25 29)(22 40 34 36)(24 26 32 30)(27 35 39 31)(28 38)(41 52 45 44)(42 55 54 51)(43 58)(46 47 50 59)(48 53)(49 56 57 60)(61 75 77 63)(62 78 66 70)(65 67 73 71)(68 76 80 72)(69 79)
G:=sub<Sym(80)| (1,58,11,48)(2,59,12,49)(3,60,13,50)(4,41,14,51)(5,42,15,52)(6,43,16,53)(7,44,17,54)(8,45,18,55)(9,46,19,56)(10,47,20,57)(21,62,31,72)(22,63,32,73)(23,64,33,74)(24,65,34,75)(25,66,35,76)(26,67,36,77)(27,68,37,78)(28,69,38,79)(29,70,39,80)(30,71,40,61), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,80)(57,79)(58,78)(59,77)(60,76), (1,6)(2,9,10,13)(3,12,19,20)(4,15,8,7)(5,18,17,14)(11,16)(21,37,25,29)(22,40,34,36)(24,26,32,30)(27,35,39,31)(28,38)(41,52,45,44)(42,55,54,51)(43,58)(46,47,50,59)(48,53)(49,56,57,60)(61,75,77,63)(62,78,66,70)(65,67,73,71)(68,76,80,72)(69,79)>;
G:=Group( (1,58,11,48)(2,59,12,49)(3,60,13,50)(4,41,14,51)(5,42,15,52)(6,43,16,53)(7,44,17,54)(8,45,18,55)(9,46,19,56)(10,47,20,57)(21,62,31,72)(22,63,32,73)(23,64,33,74)(24,65,34,75)(25,66,35,76)(26,67,36,77)(27,68,37,78)(28,69,38,79)(29,70,39,80)(30,71,40,61), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,80)(57,79)(58,78)(59,77)(60,76), (1,6)(2,9,10,13)(3,12,19,20)(4,15,8,7)(5,18,17,14)(11,16)(21,37,25,29)(22,40,34,36)(24,26,32,30)(27,35,39,31)(28,38)(41,52,45,44)(42,55,54,51)(43,58)(46,47,50,59)(48,53)(49,56,57,60)(61,75,77,63)(62,78,66,70)(65,67,73,71)(68,76,80,72)(69,79) );
G=PermutationGroup([(1,58,11,48),(2,59,12,49),(3,60,13,50),(4,41,14,51),(5,42,15,52),(6,43,16,53),(7,44,17,54),(8,45,18,55),(9,46,19,56),(10,47,20,57),(21,62,31,72),(22,63,32,73),(23,64,33,74),(24,65,34,75),(25,66,35,76),(26,67,36,77),(27,68,37,78),(28,69,38,79),(29,70,39,80),(30,71,40,61)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,40),(19,39),(20,38),(41,75),(42,74),(43,73),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,80),(57,79),(58,78),(59,77),(60,76)], [(1,6),(2,9,10,13),(3,12,19,20),(4,15,8,7),(5,18,17,14),(11,16),(21,37,25,29),(22,40,34,36),(24,26,32,30),(27,35,39,31),(28,38),(41,52,45,44),(42,55,54,51),(43,58),(46,47,50,59),(48,53),(49,56,57,60),(61,75,77,63),(62,78,66,70),(65,67,73,71),(68,76,80,72),(69,79)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 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 | 16 | 0 | 0 | 0 | 0 |
| 5 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 32 | 20 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,5,0,0,0,0,16,40,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[0,22,0,0,0,0,28,0,0,0,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,0,0,0,1,0,0,0,0,0,1,40],[32,0,0,0,0,0,20,9,0,0,0,0,0,0,1,0,40,0,0,0,0,0,40,1,0,0,0,1,40,0,0,0,0,0,40,0] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 4 | 5 | 5 | 10 | 20 | 1 | 1 | 2 | 4 | 5 | 5 | 10 | 20 | ··· | 20 | 4 | 10 | ··· | 10 | 4 | 8 | 8 | 8 | 4 | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | D4 | C4○D8 | F5 | C2×F5 | C2×F5 | C2×F5 | C22⋊F5 | C22⋊F5 | C4○D20⋊C4 |
| kernel | C4○D20⋊C4 | D20⋊C4 | Q8⋊F5 | C2×D5⋊C8 | D10.C23 | D5×C4○D4 | C4○D20 | D4⋊2D5 | Q8⋊2D5 | C5×C4○D4 | C4×D5 | C2×Dic5 | C22×D5 | D5 | C4○D4 | C2×C4 | D4 | Q8 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\circ D_{20}\rtimes C_4 % in TeX
G:=Group("C4oD20:C4"); // GroupNames label
G:=SmallGroup(320,1132);
// by ID
G=gap.SmallGroup(320,1132);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=a^2*b^9,d*b*d^-1=b^7,d*c*d^-1=a^2*b*c>;
// generators/relations