direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.D4, C4⋊C4⋊4C10, C2.7(D4×C10), C22⋊C4⋊4C10, (C22×C4)⋊3C10, (C22×C20)⋊5C2, (C2×D4).4C10, C10.70(C2×D4), (C2×C10).23D4, C22.4(C5×D4), (D4×C10).11C2, C10.43(C4○D4), (C2×C20).65C22, (C2×C10).78C23, C23.10(C2×C10), C22.13(C22×C10), (C22×C10).29C22, (C5×C4⋊C4)⋊13C2, C2.6(C5×C4○D4), (C2×C4).5(C2×C10), (C5×C22⋊C4)⋊12C2, SmallGroup(160,184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.D4
G = < a,b,c,d,e | a5=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 116 in 78 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C22.D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C5×C22.D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C22.D4, C5×D4, C22×C10, D4×C10, C5×C4○D4, C5×C22.D4
►On 80 points
Generators in S80
(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 61)(2 62)(3 63)(4 64)(5 65)(6 36)(7 37)(8 38)(9 39)(10 40)(11 51)(12 52)(13 53)(14 54)(15 55)(16 46)(17 47)(18 48)(19 49)(20 50)(21 56)(22 57)(23 58)(24 59)(25 60)(26 71)(27 72)(28 73)(29 74)(30 75)(31 66)(32 67)(33 68)(34 69)(35 70)(41 76)(42 77)(43 78)(44 79)(45 80)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 46 31 41)(2 47 32 42)(3 48 33 43)(4 49 34 44)(5 50 35 45)(6 66 11 61)(7 67 12 62)(8 68 13 63)(9 69 14 64)(10 70 15 65)(16 56 76 71)(17 57 77 72)(18 58 78 73)(19 59 79 74)(20 60 80 75)(21 51 26 36)(22 52 27 37)(23 53 28 38)(24 54 29 39)(25 55 30 40)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
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), (1,61)(2,62)(3,63)(4,64)(5,65)(6,36)(7,37)(8,38)(9,39)(10,40)(11,51)(12,52)(13,53)(14,54)(15,55)(16,46)(17,47)(18,48)(19,49)(20,50)(21,56)(22,57)(23,58)(24,59)(25,60)(26,71)(27,72)(28,73)(29,74)(30,75)(31,66)(32,67)(33,68)(34,69)(35,70)(41,76)(42,77)(43,78)(44,79)(45,80), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75), (1,46,31,41)(2,47,32,42)(3,48,33,43)(4,49,34,44)(5,50,35,45)(6,66,11,61)(7,67,12,62)(8,68,13,63)(9,69,14,64)(10,70,15,65)(16,56,76,71)(17,57,77,72)(18,58,78,73)(19,59,79,74)(20,60,80,75)(21,51,26,36)(22,52,27,37)(23,53,28,38)(24,54,29,39)(25,55,30,40), (6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)>;
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), (1,61)(2,62)(3,63)(4,64)(5,65)(6,36)(7,37)(8,38)(9,39)(10,40)(11,51)(12,52)(13,53)(14,54)(15,55)(16,46)(17,47)(18,48)(19,49)(20,50)(21,56)(22,57)(23,58)(24,59)(25,60)(26,71)(27,72)(28,73)(29,74)(30,75)(31,66)(32,67)(33,68)(34,69)(35,70)(41,76)(42,77)(43,78)(44,79)(45,80), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75), (1,46,31,41)(2,47,32,42)(3,48,33,43)(4,49,34,44)(5,50,35,45)(6,66,11,61)(7,67,12,62)(8,68,13,63)(9,69,14,64)(10,70,15,65)(16,56,76,71)(17,57,77,72)(18,58,78,73)(19,59,79,74)(20,60,80,75)(21,51,26,36)(22,52,27,37)(23,53,28,38)(24,54,29,39)(25,55,30,40), (6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75) );
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)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,36),(7,37),(8,38),(9,39),(10,40),(11,51),(12,52),(13,53),(14,54),(15,55),(16,46),(17,47),(18,48),(19,49),(20,50),(21,56),(22,57),(23,58),(24,59),(25,60),(26,71),(27,72),(28,73),(29,74),(30,75),(31,66),(32,67),(33,68),(34,69),(35,70),(41,76),(42,77),(43,78),(44,79),(45,80)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,16),(7,17),(8,18),(9,19),(10,20),(11,76),(12,77),(13,78),(14,79),(15,80),(21,31),(22,32),(23,33),(24,34),(25,35),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75)], [(1,46,31,41),(2,47,32,42),(3,48,33,43),(4,49,34,44),(5,50,35,45),(6,66,11,61),(7,67,12,62),(8,68,13,63),(9,69,14,64),(10,70,15,65),(16,56,76,71),(17,57,77,72),(18,58,78,73),(19,59,79,74),(20,60,80,75),(21,51,26,36),(22,52,27,37),(23,53,28,38),(24,54,29,39),(25,55,30,40)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,76),(17,77),(18,78),(19,79),(20,80),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75)]])
C5×C22.D4 is a maximal subgroup of
(C22×C20)⋊C4 C22⋊C4⋊D10 C10.792- 1+4 C4⋊C4.197D10 C10.802- 1+4 C10.812- 1+4 C10.1202+ 1+4 C10.1212+ 1+4 C10.822- 1+4 C4⋊C4⋊28D10 C10.612+ 1+4 C10.1222+ 1+4 C10.622+ 1+4 C10.632+ 1+4 C10.642+ 1+4 C10.842- 1+4 C10.662+ 1+4 C10.672+ 1+4 C10.852- 1+4 C10.682+ 1+4 C10.692+ 1+4
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 20A | ··· | 20P | 20Q | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C22.D4 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | D4×C10 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 4 | 12 | 8 | 4 | 4 | 2 | 4 | 8 | 16 |
Matrix representation of C5×C22.D4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 9 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,9,0,0,32,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,40,0,0,0,0,0,0,40,0,0,40,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C22.D4 in GAP, Magma, Sage, TeX
C_5\times C_2^2.D_4
% in TeX
G:=Group("C5xC2^2.D4"); // GroupNames label
G:=SmallGroup(160,184);
// by ID
G=gap.SmallGroup(160,184);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,1514,194]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations