direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C23⋊C4, C24⋊3C20, (C2×D4)⋊5C20, (D4×C10)⋊29C4, C23⋊1(C2×C20), (C22×C4)⋊3C20, (C23×C10)⋊2C4, (C22×C20)⋊18C4, C23.11(C5×D4), C24.10(C2×C10), (C22×D4).4C10, C22.10(D4×C10), (C22×C10).157D4, C22.6(C22×C20), C23.1(C22×C10), (D4×C10).282C22, (C23×C10).10C22, (C22×C10).80C23, (C2×C4)⋊1(C2×C20), (C2×C20)⋊16(C2×C4), (D4×C2×C10).16C2, (C2×C22⋊C4)⋊4C10, C22⋊C4⋊9(C2×C10), (C10×C22⋊C4)⋊9C2, (C22×C10)⋊6(C2×C4), (C2×D4).40(C2×C10), (C2×C10).405(C2×D4), C2.12(C10×C22⋊C4), C22.2(C5×C22⋊C4), C10.141(C2×C22⋊C4), (C5×C22⋊C4)⋊45C22, (C2×C10).260(C22×C4), (C2×C10).199(C22⋊C4), SmallGroup(320,910)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C23⋊C4
G = < a,b,c,d,e | a10=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 434 in 210 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C23⋊C4, C2×C22⋊C4, C22×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C23⋊C4, C5×C22⋊C4, C5×C22⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, C23×C10, C5×C23⋊C4, C10×C22⋊C4, D4×C2×C10, C10×C23⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C23⋊C4, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C2×C23⋊C4, C5×C22⋊C4, C22×C20, D4×C10, C5×C23⋊C4, C10×C22⋊C4, C10×C23⋊C4
►On 80 points
(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 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 51)(18 52)(19 53)(20 54)(21 50)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(61 78)(62 79)(63 80)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(11 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 21)(11 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 51)(10 52)(11 35 80 43)(12 36 71 44)(13 37 72 45)(14 38 73 46)(15 39 74 47)(16 40 75 48)(17 31 76 49)(18 32 77 50)(19 33 78 41)(20 34 79 42)(21 70)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)
G:=sub<Sym(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), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(61,78)(62,79)(63,80)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,21)(11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,35,80,43)(12,36,71,44)(13,37,72,45)(14,38,73,46)(15,39,74,47)(16,40,75,48)(17,31,76,49)(18,32,77,50)(19,33,78,41)(20,34,79,42)(21,70)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)>;
G:=Group( (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), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(61,78)(62,79)(63,80)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,21)(11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,35,80,43)(12,36,71,44)(13,37,72,45)(14,38,73,46)(15,39,74,47)(16,40,75,48)(17,31,76,49)(18,32,77,50)(19,33,78,41)(20,34,79,42)(21,70)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69) );
G=PermutationGroup([[(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)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,31),(10,32),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,51),(18,52),(19,53),(20,54),(21,50),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(61,78),(62,79),(63,80),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77)], [(11,80),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(51,69),(52,70),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,21),(11,80),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(31,49),(32,50),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(51,69),(52,70),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,51),(10,52),(11,35,80,43),(12,36,71,44),(13,37,72,45),(14,38,73,46),(15,39,74,47),(16,40,75,48),(17,31,76,49),(18,32,77,50),(19,33,78,41),(20,34,79,42),(21,70),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | ··· | 4J | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 10AK | ··· | 10AR | 20A | ··· | 20AN |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | C20 | D4 | C5×D4 | C23⋊C4 | C5×C23⋊C4 |
| kernel | C10×C23⋊C4 | C5×C23⋊C4 | C10×C22⋊C4 | D4×C2×C10 | C22×C20 | D4×C10 | C23×C10 | C2×C23⋊C4 | C23⋊C4 | C2×C22⋊C4 | C22×D4 | C22×C4 | C2×D4 | C24 | C22×C10 | C23 | C10 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 4 | 2 | 4 | 16 | 8 | 4 | 8 | 16 | 8 | 4 | 16 | 2 | 8 |
Matrix representation of C10×C23⋊C4 ►in GL6(𝔽41)
| 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [25,0,0,0,0,0,0,25,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
C10×C23⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times C_2^3\rtimes C_4 % in TeX
G:=Group("C10xC2^3:C4"); // GroupNames label
G:=SmallGroup(320,910);
// by ID
G=gap.SmallGroup(320,910);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,5052]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^2=e^4=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