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 [×6], C2 [×4], C4 [×4], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×4], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×2], C24, C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C5×C22⋊C4 [×4], C22×C20 [×2], C23×C10, C10×C22⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], 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 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 71)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 61)(10 62)(11 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 58)(42 59)(43 60)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(50 57)
(1 28 48 20)(2 29 49 11)(3 30 50 12)(4 21 41 13)(5 22 42 14)(6 23 43 15)(7 24 44 16)(8 25 45 17)(9 26 46 18)(10 27 47 19)(31 56 80 64)(32 57 71 65)(33 58 72 66)(34 59 73 67)(35 60 74 68)(36 51 75 69)(37 52 76 70)(38 53 77 61)(39 54 78 62)(40 55 79 63)
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)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,71)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,61)(10,62)(11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,58)(42,59)(43,60)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57), (1,28,48,20)(2,29,49,11)(3,30,50,12)(4,21,41,13)(5,22,42,14)(6,23,43,15)(7,24,44,16)(8,25,45,17)(9,26,46,18)(10,27,47,19)(31,56,80,64)(32,57,71,65)(33,58,72,66)(34,59,73,67)(35,60,74,68)(36,51,75,69)(37,52,76,70)(38,53,77,61)(39,54,78,62)(40,55,79,63)>;
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)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,71)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,61)(10,62)(11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,58)(42,59)(43,60)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57), (1,28,48,20)(2,29,49,11)(3,30,50,12)(4,21,41,13)(5,22,42,14)(6,23,43,15)(7,24,44,16)(8,25,45,17)(9,26,46,18)(10,27,47,19)(31,56,80,64)(32,57,71,65)(33,58,72,66)(34,59,73,67)(35,60,74,68)(36,51,75,69)(37,52,76,70)(38,53,77,61)(39,54,78,62)(40,55,79,63) );
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),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,71),(51,69),(52,70),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,61),(10,62),(11,80),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,31),(30,32),(41,58),(42,59),(43,60),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(50,57)], [(1,28,48,20),(2,29,49,11),(3,30,50,12),(4,21,41,13),(5,22,42,14),(6,23,43,15),(7,24,44,16),(8,25,45,17),(9,26,46,18),(10,27,47,19),(31,56,80,64),(32,57,71,65),(33,58,72,66),(34,59,73,67),(35,60,74,68),(36,51,75,69),(37,52,76,70),(38,53,77,61),(39,54,78,62),(40,55,79,63)])
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