direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22⋊C4, C24.C10, C23⋊2C20, C2.1(D4×C10), C22⋊2(C2×C20), (C22×C10)⋊6C4, (C22×C4)⋊1C10, (C22×C20)⋊3C2, (C2×C10).50D4, C10.64(C2×D4), (C2×C20)⋊11C22, C23.5(C2×C10), C2.1(C22×C20), (C23×C10).1C2, C22.12(C5×D4), C10.42(C22×C4), (C2×C10).70C23, C22.4(C22×C10), (C22×C10).24C22, (C2×C4)⋊3(C2×C10), (C2×C10)⋊10(C2×C4), SmallGroup(160,176)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C22⋊C4
G = < a,b,c,d | a10=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 188 in 132 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C5×C22⋊C4, C22×C20, C23×C10, C10×C22⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊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 48)(2 49)(3 50)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 79)(12 80)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
(1 67)(2 68)(3 69)(4 70)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 40)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 79)(22 80)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)
(1 11 48 21)(2 12 49 22)(3 13 50 23)(4 14 41 24)(5 15 42 25)(6 16 43 26)(7 17 44 27)(8 18 45 28)(9 19 46 29)(10 20 47 30)(31 52 80 68)(32 53 71 69)(33 54 72 70)(34 55 73 61)(35 56 74 62)(36 57 75 63)(37 58 76 64)(38 59 77 65)(39 60 78 66)(40 51 79 67)
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), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,79)(12,80)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,40)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,79)(22,80)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53), (1,11,48,21)(2,12,49,22)(3,13,50,23)(4,14,41,24)(5,15,42,25)(6,16,43,26)(7,17,44,27)(8,18,45,28)(9,19,46,29)(10,20,47,30)(31,52,80,68)(32,53,71,69)(33,54,72,70)(34,55,73,61)(35,56,74,62)(36,57,75,63)(37,58,76,64)(38,59,77,65)(39,60,78,66)(40,51,79,67)>;
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), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,79)(12,80)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,40)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,79)(22,80)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53), (1,11,48,21)(2,12,49,22)(3,13,50,23)(4,14,41,24)(5,15,42,25)(6,16,43,26)(7,17,44,27)(8,18,45,28)(9,19,46,29)(10,20,47,30)(31,52,80,68)(32,53,71,69)(33,54,72,70)(34,55,73,61)(35,56,74,62)(36,57,75,63)(37,58,76,64)(38,59,77,65)(39,60,78,66)(40,51,79,67) );
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)], [(1,48),(2,49),(3,50),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,79),(12,80),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,40),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(51,67),(52,68),(53,69),(54,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66)], [(1,67),(2,68),(3,69),(4,70),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,40),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,79),(22,80),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,51),(49,52),(50,53)], [(1,11,48,21),(2,12,49,22),(3,13,50,23),(4,14,41,24),(5,15,42,25),(6,16,43,26),(7,17,44,27),(8,18,45,28),(9,19,46,29),(10,20,47,30),(31,52,80,68),(32,53,71,69),(33,54,72,70),(34,55,73,61),(35,56,74,62),(36,57,75,63),(37,58,76,64),(38,59,77,65),(39,60,78,66),(40,51,79,67)]])
C10×C22⋊C4 is a maximal subgroup of
C24.Dic5 C24.D10 C24.2D10 C24.44D10 C23.42D20 C24.3D10 C24.4D10 C24.46D10 C23⋊Dic10 C24.6D10 C24.7D10 C24.47D10 C24.8D10 C24.9D10 C23.14D20 C24.48D10 C24.12D10 C24.13D10 C23.45D20 C24.14D10 C23⋊2D20 C24.16D10 C23⋊2Dic10 C24.24D10 C24.27D10 C23⋊3D20 C24.30D10 C24.31D10 D4×C2×C20
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D4 | C5×D4 |
| kernel | C10×C22⋊C4 | C5×C22⋊C4 | C22×C20 | C23×C10 | C22×C10 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C2×C10 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 16 | 8 | 4 | 32 | 4 | 16 |
Matrix representation of C10×C22⋊C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0] >;
C10×C22⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times C_2^2\rtimes C_4 % in TeX
G:=Group("C10xC2^2:C4"); // GroupNames label
G:=SmallGroup(160,176);
// by ID
G=gap.SmallGroup(160,176);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations