direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C23.D4, C22⋊C4⋊2C20, (C22×C20)⋊4C4, (C22×C4)⋊2C20, (C2×C20).18D4, C4.D4.C10, C23.2(C5×D4), C23⋊C4.1C10, C23.2(C2×C20), (C22×C10).2D4, C10.54(C23⋊C4), (D4×C10).175C22, C22.D4.1C10, (C2×C4).2(C5×D4), C2.7(C5×C23⋊C4), (C5×C22⋊C4)⋊10C4, (C2×D4).2(C2×C10), (C5×C23⋊C4).3C2, (C5×C4.D4).2C2, (C22×C10).34(C2×C4), C22.11(C5×C22⋊C4), (C5×C22.D4).4C2, (C2×C10).138(C22⋊C4), SmallGroup(320,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be3 >
Subgroups: 162 in 68 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, C20, C2×C10, C2×C10, C23⋊C4, C4.D4, C22.D4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C23.D4, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×M4(2), C22×C20, D4×C10, C5×C23⋊C4, C5×C4.D4, C5×C22.D4, C5×C23.D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, C2×C10, C23⋊C4, C2×C20, C5×D4, C23.D4, C5×C22⋊C4, C5×C23⋊C4, C5×C23.D4
►On 80 points
Generators in S80
(1 43 73 34 65)(2 44 74 35 66)(3 45 75 36 67)(4 46 76 37 68)(5 47 77 38 69)(6 48 78 39 70)(7 41 79 40 71)(8 42 80 33 72)(9 51 64 26 18)(10 52 57 27 19)(11 53 58 28 20)(12 54 59 29 21)(13 55 60 30 22)(14 56 61 31 23)(15 49 62 32 24)(16 50 63 25 17)
(2 54)(3 7)(4 52)(6 50)(8 56)(9 13)(10 68)(12 66)(14 72)(16 70)(17 39)(18 22)(19 37)(21 35)(23 33)(25 78)(26 30)(27 76)(29 74)(31 80)(36 40)(41 45)(42 61)(44 59)(46 57)(48 63)(51 55)(60 64)(67 71)(75 79)
(1 49)(2 54)(3 51)(4 56)(5 53)(6 50)(7 55)(8 52)(9 67)(10 72)(11 69)(12 66)(13 71)(14 68)(15 65)(16 70)(17 39)(18 36)(19 33)(20 38)(21 35)(22 40)(23 37)(24 34)(25 78)(26 75)(27 80)(28 77)(29 74)(30 79)(31 76)(32 73)(41 60)(42 57)(43 62)(44 59)(45 64)(46 61)(47 58)(48 63)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 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)
(2 52 54 4)(3 51 7 55)(6 56 50 8)(9 71 13 67)(10 12 68 66)(11 15)(14 16 72 70)(17 33 39 23)(18 40 22 36)(19 21 37 35)(20 24)(25 80 78 31)(26 79 30 75)(27 29 76 74)(28 32)(41 60 45 64)(42 48 61 63)(44 57 59 46)(49 53)(58 62)
G:=sub<Sym(80)| (1,43,73,34,65)(2,44,74,35,66)(3,45,75,36,67)(4,46,76,37,68)(5,47,77,38,69)(6,48,78,39,70)(7,41,79,40,71)(8,42,80,33,72)(9,51,64,26,18)(10,52,57,27,19)(11,53,58,28,20)(12,54,59,29,21)(13,55,60,30,22)(14,56,61,31,23)(15,49,62,32,24)(16,50,63,25,17), (2,54)(3,7)(4,52)(6,50)(8,56)(9,13)(10,68)(12,66)(14,72)(16,70)(17,39)(18,22)(19,37)(21,35)(23,33)(25,78)(26,30)(27,76)(29,74)(31,80)(36,40)(41,45)(42,61)(44,59)(46,57)(48,63)(51,55)(60,64)(67,71)(75,79), (1,49)(2,54)(3,51)(4,56)(5,53)(6,50)(7,55)(8,52)(9,67)(10,72)(11,69)(12,66)(13,71)(14,68)(15,65)(16,70)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34)(25,78)(26,75)(27,80)(28,77)(29,74)(30,79)(31,76)(32,73)(41,60)(42,57)(43,62)(44,59)(45,64)(46,61)(47,58)(48,63), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (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), (2,52,54,4)(3,51,7,55)(6,56,50,8)(9,71,13,67)(10,12,68,66)(11,15)(14,16,72,70)(17,33,39,23)(18,40,22,36)(19,21,37,35)(20,24)(25,80,78,31)(26,79,30,75)(27,29,76,74)(28,32)(41,60,45,64)(42,48,61,63)(44,57,59,46)(49,53)(58,62)>;
G:=Group( (1,43,73,34,65)(2,44,74,35,66)(3,45,75,36,67)(4,46,76,37,68)(5,47,77,38,69)(6,48,78,39,70)(7,41,79,40,71)(8,42,80,33,72)(9,51,64,26,18)(10,52,57,27,19)(11,53,58,28,20)(12,54,59,29,21)(13,55,60,30,22)(14,56,61,31,23)(15,49,62,32,24)(16,50,63,25,17), (2,54)(3,7)(4,52)(6,50)(8,56)(9,13)(10,68)(12,66)(14,72)(16,70)(17,39)(18,22)(19,37)(21,35)(23,33)(25,78)(26,30)(27,76)(29,74)(31,80)(36,40)(41,45)(42,61)(44,59)(46,57)(48,63)(51,55)(60,64)(67,71)(75,79), (1,49)(2,54)(3,51)(4,56)(5,53)(6,50)(7,55)(8,52)(9,67)(10,72)(11,69)(12,66)(13,71)(14,68)(15,65)(16,70)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34)(25,78)(26,75)(27,80)(28,77)(29,74)(30,79)(31,76)(32,73)(41,60)(42,57)(43,62)(44,59)(45,64)(46,61)(47,58)(48,63), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (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), (2,52,54,4)(3,51,7,55)(6,56,50,8)(9,71,13,67)(10,12,68,66)(11,15)(14,16,72,70)(17,33,39,23)(18,40,22,36)(19,21,37,35)(20,24)(25,80,78,31)(26,79,30,75)(27,29,76,74)(28,32)(41,60,45,64)(42,48,61,63)(44,57,59,46)(49,53)(58,62) );
G=PermutationGroup([[(1,43,73,34,65),(2,44,74,35,66),(3,45,75,36,67),(4,46,76,37,68),(5,47,77,38,69),(6,48,78,39,70),(7,41,79,40,71),(8,42,80,33,72),(9,51,64,26,18),(10,52,57,27,19),(11,53,58,28,20),(12,54,59,29,21),(13,55,60,30,22),(14,56,61,31,23),(15,49,62,32,24),(16,50,63,25,17)], [(2,54),(3,7),(4,52),(6,50),(8,56),(9,13),(10,68),(12,66),(14,72),(16,70),(17,39),(18,22),(19,37),(21,35),(23,33),(25,78),(26,30),(27,76),(29,74),(31,80),(36,40),(41,45),(42,61),(44,59),(46,57),(48,63),(51,55),(60,64),(67,71),(75,79)], [(1,49),(2,54),(3,51),(4,56),(5,53),(6,50),(7,55),(8,52),(9,67),(10,72),(11,69),(12,66),(13,71),(14,68),(15,65),(16,70),(17,39),(18,36),(19,33),(20,38),(21,35),(22,40),(23,37),(24,34),(25,78),(26,75),(27,80),(28,77),(29,74),(30,79),(31,76),(32,73),(41,60),(42,57),(43,62),(44,59),(45,64),(46,61),(47,58),(48,63)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80)], [(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)], [(2,52,54,4),(3,51,7,55),(6,56,50,8),(9,71,13,67),(10,12,68,66),(11,15),(14,16,72,70),(17,33,39,23),(18,40,22,36),(19,21,37,35),(20,24),(25,80,78,31),(26,79,30,75),(27,29,76,74),(28,32),(41,60,45,64),(42,48,61,63),(44,57,59,46),(49,53),(58,62)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 20A | ··· | 20L | 20M | ··· | 20X | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | D4 | D4 | C5×D4 | C5×D4 | C23⋊C4 | C23.D4 | C5×C23⋊C4 | C5×C23.D4 |
| kernel | C5×C23.D4 | C5×C23⋊C4 | C5×C4.D4 | C5×C22.D4 | C5×C22⋊C4 | C22×C20 | C23.D4 | C23⋊C4 | C4.D4 | C22.D4 | C22⋊C4 | C22×C4 | C2×C20 | C22×C10 | C2×C4 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 4 | 4 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C23.D4 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 16 | 16 | 16 | 25 |
| 25 | 25 | 16 | 25 |
| 16 | 25 | 25 | 25 |
| 16 | 25 | 16 | 16 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,0,40,0,0,40,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[16,25,16,16,16,25,25,25,16,16,25,16,25,25,25,16],[0,0,1,0,0,0,0,40,0,1,0,0,40,0,0,0] >;
C5×C23.D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3.D_4
% in TeX
G:=Group("C5xC2^3.D4"); // GroupNames label
G:=SmallGroup(320,157);
// by ID
G=gap.SmallGroup(320,157);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,2111,375,10085]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^3>;
// generators/relations