direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.37D4, (C2×D4)⋊8C20, (D4×C10)⋊32C4, D4.6(C2×C20), C4.55(D4×C10), (C2×C40)⋊38C22, C20.462(C2×D4), (C2×C20).315D4, C4.5(C22×C20), D4⋊C4⋊15C10, C23.37(C5×D4), C42⋊C2⋊3C10, (C22×D4).7C10, C22.45(D4×C10), (C2×M4(2))⋊11C10, (C10×M4(2))⋊29C2, C20.209(C22×C4), (C2×C20).894C23, (C22×C10).159D4, C10.127(C8⋊C22), C20.129(C22⋊C4), (D4×C10).289C22, (C22×C20).411C22, C4⋊C4⋊8(C2×C10), (C2×C8)⋊8(C2×C10), (D4×C2×C10).19C2, (C2×C4).23(C5×D4), C2.2(C5×C8⋊C22), (C5×C4⋊C4)⋊64C22, (C2×C4).21(C2×C20), (C5×D4).42(C2×C4), C4.14(C5×C22⋊C4), (C5×D4⋊C4)⋊38C2, (C2×C20).367(C2×C4), (C2×D4).47(C2×C10), (C2×C10).621(C2×D4), C2.21(C10×C22⋊C4), (C5×C42⋊C2)⋊24C2, C10.150(C2×C22⋊C4), (C22×C4).30(C2×C10), (C2×C4).69(C22×C10), C22.20(C5×C22⋊C4), (C2×C10).145(C22⋊C4), SmallGroup(320,919)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.37D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 386 in 190 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C20, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, C23.37D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C22×C20, D4×C10, D4×C10, C23×C10, C5×D4⋊C4, C5×C42⋊C2, C10×M4(2), D4×C2×C10, C5×C23.37D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C8⋊C22, C2×C20, C5×D4, C22×C10, C23.37D4, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C8⋊C22, C5×C23.37D4
►On 80 points
Generators in S80
(1 58 23 50 15)(2 59 24 51 16)(3 60 17 52 9)(4 61 18 53 10)(5 62 19 54 11)(6 63 20 55 12)(7 64 21 56 13)(8 57 22 49 14)(25 42 76 33 68)(26 43 77 34 69)(27 44 78 35 70)(28 45 79 36 71)(29 46 80 37 72)(30 47 73 38 65)(31 48 74 39 66)(32 41 75 40 67)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 69)(10 66)(11 71)(12 68)(13 65)(14 70)(15 67)(16 72)(17 77)(18 74)(19 79)(20 76)(21 73)(22 78)(23 75)(24 80)(33 55)(34 52)(35 49)(36 54)(37 51)(38 56)(39 53)(40 50)(41 58)(42 63)(43 60)(44 57)(45 62)(46 59)(47 64)(48 61)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(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)
(1 8 28 27)(2 26 29 7)(3 6 30 25)(4 32 31 5)(9 12 65 68)(10 67 66 11)(13 16 69 72)(14 71 70 15)(17 20 73 76)(18 75 74 19)(21 24 77 80)(22 79 78 23)(33 52 55 38)(34 37 56 51)(35 50 49 36)(39 54 53 40)(41 48 62 61)(42 60 63 47)(43 46 64 59)(44 58 57 45)
G:=sub<Sym(80)| (1,58,23,50,15)(2,59,24,51,16)(3,60,17,52,9)(4,61,18,53,10)(5,62,19,54,11)(6,63,20,55,12)(7,64,21,56,13)(8,57,22,49,14)(25,42,76,33,68)(26,43,77,34,69)(27,44,78,35,70)(28,45,79,36,71)(29,46,80,37,72)(30,47,73,38,65)(31,48,74,39,66)(32,41,75,40,67), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,69)(10,66)(11,71)(12,68)(13,65)(14,70)(15,67)(16,72)(17,77)(18,74)(19,79)(20,76)(21,73)(22,78)(23,75)(24,80)(33,55)(34,52)(35,49)(36,54)(37,51)(38,56)(39,53)(40,50)(41,58)(42,63)(43,60)(44,57)(45,62)(46,59)(47,64)(48,61), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (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), (1,8,28,27)(2,26,29,7)(3,6,30,25)(4,32,31,5)(9,12,65,68)(10,67,66,11)(13,16,69,72)(14,71,70,15)(17,20,73,76)(18,75,74,19)(21,24,77,80)(22,79,78,23)(33,52,55,38)(34,37,56,51)(35,50,49,36)(39,54,53,40)(41,48,62,61)(42,60,63,47)(43,46,64,59)(44,58,57,45)>;
G:=Group( (1,58,23,50,15)(2,59,24,51,16)(3,60,17,52,9)(4,61,18,53,10)(5,62,19,54,11)(6,63,20,55,12)(7,64,21,56,13)(8,57,22,49,14)(25,42,76,33,68)(26,43,77,34,69)(27,44,78,35,70)(28,45,79,36,71)(29,46,80,37,72)(30,47,73,38,65)(31,48,74,39,66)(32,41,75,40,67), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,69)(10,66)(11,71)(12,68)(13,65)(14,70)(15,67)(16,72)(17,77)(18,74)(19,79)(20,76)(21,73)(22,78)(23,75)(24,80)(33,55)(34,52)(35,49)(36,54)(37,51)(38,56)(39,53)(40,50)(41,58)(42,63)(43,60)(44,57)(45,62)(46,59)(47,64)(48,61), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (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), (1,8,28,27)(2,26,29,7)(3,6,30,25)(4,32,31,5)(9,12,65,68)(10,67,66,11)(13,16,69,72)(14,71,70,15)(17,20,73,76)(18,75,74,19)(21,24,77,80)(22,79,78,23)(33,52,55,38)(34,37,56,51)(35,50,49,36)(39,54,53,40)(41,48,62,61)(42,60,63,47)(43,46,64,59)(44,58,57,45) );
G=PermutationGroup([[(1,58,23,50,15),(2,59,24,51,16),(3,60,17,52,9),(4,61,18,53,10),(5,62,19,54,11),(6,63,20,55,12),(7,64,21,56,13),(8,57,22,49,14),(25,42,76,33,68),(26,43,77,34,69),(27,44,78,35,70),(28,45,79,36,71),(29,46,80,37,72),(30,47,73,38,65),(31,48,74,39,66),(32,41,75,40,67)], [(1,32),(2,29),(3,26),(4,31),(5,28),(6,25),(7,30),(8,27),(9,69),(10,66),(11,71),(12,68),(13,65),(14,70),(15,67),(16,72),(17,77),(18,74),(19,79),(20,76),(21,73),(22,78),(23,75),(24,80),(33,55),(34,52),(35,49),(36,54),(37,51),(38,56),(39,53),(40,50),(41,58),(42,63),(43,60),(44,57),(45,62),(46,59),(47,64),(48,61)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(33,55),(34,56),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,62),(42,63),(43,64),(44,57),(45,58),(46,59),(47,60),(48,61)], [(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)], [(1,8,28,27),(2,26,29,7),(3,6,30,25),(4,32,31,5),(9,12,65,68),(10,67,66,11),(13,16,69,72),(14,71,70,15),(17,20,73,76),(18,75,74,19),(21,24,77,80),(22,79,78,23),(33,52,55,38),(34,37,56,51),(35,50,49,36),(39,54,53,40),(41,48,62,61),(42,60,63,47),(43,46,64,59),(44,58,57,45)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 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 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | C5×D4 | C5×D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C23.37D4 | C5×D4⋊C4 | C5×C42⋊C2 | C10×M4(2) | D4×C2×C10 | D4×C10 | C23.37D4 | D4⋊C4 | C42⋊C2 | C2×M4(2) | C22×D4 | C2×D4 | C2×C20 | C22×C10 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 3 | 1 | 12 | 4 | 2 | 8 |
Matrix representation of C5×C23.37D4 ►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 |
| 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 | 28 | 0 | 0 | 1 |
| 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 | 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 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 0 | 40 | 40 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 40 | 27 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 14 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 27 | 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,28,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,0,27,1,0,0,0,0,0,2,40,0,0,1,40,0,27,0,0,0,40,0,0],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,14,1,0,0,0,0,0,0,1,0,0,1,0,0,27,0,0,0,1,0,0] >;
C5×C23.37D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{37}D_4 % in TeX
G:=Group("C5xC2^3.37D4"); // GroupNames label
G:=SmallGroup(320,919);
// by ID
G=gap.SmallGroup(320,919);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1731,856,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-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=c*d*e^3>;
// generators/relations