direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23⋊C8, C23⋊C40, C24.1C20, C22⋊C8⋊1C10, (C22×C10)⋊1C8, (C2×C20).441D4, C22.2(C2×C40), (C23×C10).1C4, (C22×C4).1C20, (C22×C20).4C4, C23.21(C2×C20), C10.35(C22⋊C8), C10.48(C23⋊C4), (C2×C10).32M4(2), (C22×C20).1C22, C22.2(C5×M4(2)), C10.18(C4.D4), (C5×C22⋊C8)⋊3C2, (C2×C4).91(C5×D4), C2.3(C5×C22⋊C8), C2.1(C5×C23⋊C4), (C2×C10).48(C2×C8), C2.1(C5×C4.D4), (C2×C22⋊C4).1C10, (C10×C22⋊C4).3C2, (C22×C4).1(C2×C10), C22.23(C5×C22⋊C4), (C22×C10).174(C2×C4), (C2×C10).182(C22⋊C4), SmallGroup(320,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23⋊C8
G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 218 in 98 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, C23, C10, C10, C22⋊C4, C2×C8, C22×C4, C24, C20, C2×C10, C2×C10, C22⋊C8, C2×C22⋊C4, C40, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23⋊C8, C5×C22⋊C4, C2×C40, C22×C20, C23×C10, C5×C22⋊C8, C10×C22⋊C4, C5×C23⋊C8
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D4, C10, C22⋊C4, C2×C8, M4(2), C20, C2×C10, C22⋊C8, C23⋊C4, C4.D4, C40, C2×C20, C5×D4, C23⋊C8, C5×C22⋊C4, C2×C40, C5×M4(2), C5×C22⋊C8, C5×C23⋊C4, C5×C4.D4, C5×C23⋊C8
►On 80 points
Generators in S80
(1 53 10 57 17)(2 54 11 58 18)(3 55 12 59 19)(4 56 13 60 20)(5 49 14 61 21)(6 50 15 62 22)(7 51 16 63 23)(8 52 9 64 24)(25 47 73 33 65)(26 48 74 34 66)(27 41 75 35 67)(28 42 76 36 68)(29 43 77 37 69)(30 44 78 38 70)(31 45 79 39 71)(32 46 80 40 72)
(2 28)(3 25)(4 8)(6 32)(7 29)(9 13)(11 76)(12 73)(15 80)(16 77)(18 68)(19 65)(20 24)(22 72)(23 69)(26 30)(33 59)(34 38)(36 58)(37 63)(40 62)(42 54)(43 51)(44 48)(46 50)(47 55)(52 56)(60 64)(66 70)(74 78)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 74)(10 14)(11 76)(12 16)(13 78)(15 80)(17 21)(18 68)(19 23)(20 70)(22 72)(24 66)(25 29)(27 31)(33 37)(34 64)(35 39)(36 58)(38 60)(40 62)(41 45)(42 54)(43 47)(44 56)(46 50)(48 52)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 78)(10 79)(11 80)(12 73)(13 74)(14 75)(15 76)(16 77)(17 71)(18 72)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(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)
G:=sub<Sym(80)| (1,53,10,57,17)(2,54,11,58,18)(3,55,12,59,19)(4,56,13,60,20)(5,49,14,61,21)(6,50,15,62,22)(7,51,16,63,23)(8,52,9,64,24)(25,47,73,33,65)(26,48,74,34,66)(27,41,75,35,67)(28,42,76,36,68)(29,43,77,37,69)(30,44,78,38,70)(31,45,79,39,71)(32,46,80,40,72), (2,28)(3,25)(4,8)(6,32)(7,29)(9,13)(11,76)(12,73)(15,80)(16,77)(18,68)(19,65)(20,24)(22,72)(23,69)(26,30)(33,59)(34,38)(36,58)(37,63)(40,62)(42,54)(43,51)(44,48)(46,50)(47,55)(52,56)(60,64)(66,70)(74,78), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,74)(10,14)(11,76)(12,16)(13,78)(15,80)(17,21)(18,68)(19,23)(20,70)(22,72)(24,66)(25,29)(27,31)(33,37)(34,64)(35,39)(36,58)(38,60)(40,62)(41,45)(42,54)(43,47)(44,56)(46,50)(48,52)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,78)(10,79)(11,80)(12,73)(13,74)(14,75)(15,76)(16,77)(17,71)(18,72)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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)>;
G:=Group( (1,53,10,57,17)(2,54,11,58,18)(3,55,12,59,19)(4,56,13,60,20)(5,49,14,61,21)(6,50,15,62,22)(7,51,16,63,23)(8,52,9,64,24)(25,47,73,33,65)(26,48,74,34,66)(27,41,75,35,67)(28,42,76,36,68)(29,43,77,37,69)(30,44,78,38,70)(31,45,79,39,71)(32,46,80,40,72), (2,28)(3,25)(4,8)(6,32)(7,29)(9,13)(11,76)(12,73)(15,80)(16,77)(18,68)(19,65)(20,24)(22,72)(23,69)(26,30)(33,59)(34,38)(36,58)(37,63)(40,62)(42,54)(43,51)(44,48)(46,50)(47,55)(52,56)(60,64)(66,70)(74,78), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,74)(10,14)(11,76)(12,16)(13,78)(15,80)(17,21)(18,68)(19,23)(20,70)(22,72)(24,66)(25,29)(27,31)(33,37)(34,64)(35,39)(36,58)(38,60)(40,62)(41,45)(42,54)(43,47)(44,56)(46,50)(48,52)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,78)(10,79)(11,80)(12,73)(13,74)(14,75)(15,76)(16,77)(17,71)(18,72)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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) );
G=PermutationGroup([[(1,53,10,57,17),(2,54,11,58,18),(3,55,12,59,19),(4,56,13,60,20),(5,49,14,61,21),(6,50,15,62,22),(7,51,16,63,23),(8,52,9,64,24),(25,47,73,33,65),(26,48,74,34,66),(27,41,75,35,67),(28,42,76,36,68),(29,43,77,37,69),(30,44,78,38,70),(31,45,79,39,71),(32,46,80,40,72)], [(2,28),(3,25),(4,8),(6,32),(7,29),(9,13),(11,76),(12,73),(15,80),(16,77),(18,68),(19,65),(20,24),(22,72),(23,69),(26,30),(33,59),(34,38),(36,58),(37,63),(40,62),(42,54),(43,51),(44,48),(46,50),(47,55),(52,56),(60,64),(66,70),(74,78)], [(1,5),(2,28),(3,7),(4,30),(6,32),(8,26),(9,74),(10,14),(11,76),(12,16),(13,78),(15,80),(17,21),(18,68),(19,23),(20,70),(22,72),(24,66),(25,29),(27,31),(33,37),(34,64),(35,39),(36,58),(38,60),(40,62),(41,45),(42,54),(43,47),(44,56),(46,50),(48,52),(49,53),(51,55),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,78),(10,79),(11,80),(12,73),(13,74),(14,75),(15,76),(16,77),(17,71),(18,72),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,57),(40,58),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(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)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 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 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 | D4 | M4(2) | C5×D4 | C5×M4(2) | C23⋊C4 | C4.D4 | C5×C23⋊C4 | C5×C4.D4 |
| kernel | C5×C23⋊C8 | C5×C22⋊C8 | C10×C22⋊C4 | C22×C20 | C23×C10 | C23⋊C8 | C22×C10 | C22⋊C8 | C2×C22⋊C4 | C22×C4 | C24 | C23 | C2×C20 | C2×C10 | C2×C4 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 8 | 32 | 2 | 2 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C23⋊C8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 24 | 40 | 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 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 |
| 24 | 39 | 0 | 0 | 0 | 0 |
| 17 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[24,17,0,0,0,0,39,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C5×C23⋊C8 in GAP, Magma, Sage, TeX
C_5\times C_2^3\rtimes C_8
% in TeX
G:=Group("C5xC2^3:C8"); // GroupNames label
G:=SmallGroup(320,128);
// by ID
G=gap.SmallGroup(320,128);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations