direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C22⋊F5, D10⋊4C42, D5.4(C4×D4), C22⋊1(C4×F5), (C22×C4)⋊6F5, (C2×C10)⋊2C42, C20⋊3(C22⋊C4), (C22×C20)⋊15C4, (C4×D5).120D4, D10.91(C2×D4), C23.48(C2×F5), C10.20(C2×C42), D10.3Q8⋊9C2, D10.25(C4○D4), Dic5⋊5(C22⋊C4), (C22×Dic5)⋊17C4, D10.37(C22×C4), C22.50(C22×F5), C10.17(C42⋊C2), (C22×F5).17C22, (C23×D5).133C22, (C22×D5).275C23, C2.7(D10.C23), (C2×C4×F5)⋊9C2, (C2×C4×D5)⋊17C4, C5⋊2(C4×C22⋊C4), C2.20(C2×C4×F5), (C2×F5)⋊2(C2×C4), C2.4(C2×C22⋊F5), (C2×C4).171(C2×F5), (D5×C22×C4).35C2, (C2×C22⋊F5).8C2, (C2×C20).179(C2×C4), C10.10(C2×C22⋊C4), (C2×C4×D5).366C22, (C2×C10).70(C22×C4), (C22×C10).70(C2×C4), (C22×D5).90(C2×C4), (C2×Dic5).189(C2×C4), SmallGroup(320,1101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C22⋊F5
G = < a,b,c,d,e | a4=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 1002 in 258 conjugacy classes, 86 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C2×C10, C2×C10, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C4×C22⋊C4, C4×F5, C22⋊F5, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C22×F5, C23×D5, D10.3Q8, C2×C4×F5, C2×C22⋊F5, D5×C22×C4, C4×C22⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, F5, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C2×F5, C4×C22⋊C4, C4×F5, C22⋊F5, C22×F5, C2×C4×F5, D10.C23, C2×C22⋊F5, C4×C22⋊F5
►On 80 points
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(1 54)(2 55)(3 51)(4 52)(5 53)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 14)(2 11 5 12)(3 13 4 15)(6 18 7 20)(8 17 10 16)(9 19)(21 33 22 35)(23 32 25 31)(24 34)(26 38 27 40)(28 37 30 36)(29 39)(41 58 42 60)(43 57 45 56)(44 59)(46 53 47 55)(48 52 50 51)(49 54)(61 78 62 80)(63 77 65 76)(64 79)(66 73 67 75)(68 72 70 71)(69 74)
G:=sub<Sym(80)| (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,54)(2,55)(3,51)(4,52)(5,53)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,14)(2,11,5,12)(3,13,4,15)(6,18,7,20)(8,17,10,16)(9,19)(21,33,22,35)(23,32,25,31)(24,34)(26,38,27,40)(28,37,30,36)(29,39)(41,58,42,60)(43,57,45,56)(44,59)(46,53,47,55)(48,52,50,51)(49,54)(61,78,62,80)(63,77,65,76)(64,79)(66,73,67,75)(68,72,70,71)(69,74)>;
G:=Group( (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,54)(2,55)(3,51)(4,52)(5,53)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,14)(2,11,5,12)(3,13,4,15)(6,18,7,20)(8,17,10,16)(9,19)(21,33,22,35)(23,32,25,31)(24,34)(26,38,27,40)(28,37,30,36)(29,39)(41,58,42,60)(43,57,45,56)(44,59)(46,53,47,55)(48,52,50,51)(49,54)(61,78,62,80)(63,77,65,76)(64,79)(66,73,67,75)(68,72,70,71)(69,74) );
G=PermutationGroup([[(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70)], [(1,54),(2,55),(3,51),(4,52),(5,53),(6,56),(7,57),(8,58),(9,59),(10,60),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(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)], [(1,14),(2,11,5,12),(3,13,4,15),(6,18,7,20),(8,17,10,16),(9,19),(21,33,22,35),(23,32,25,31),(24,34),(26,38,27,40),(28,37,30,36),(29,39),(41,58,42,60),(43,57,45,56),(44,59),(46,53,47,55),(48,52,50,51),(49,54),(61,78,62,80),(63,77,65,76),(64,79),(66,73,67,75),(68,72,70,71),(69,74)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4AB | 5 | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | F5 | C2×F5 | C2×F5 | C22⋊F5 | C4×F5 | D10.C23 |
| kernel | C4×C22⋊F5 | D10.3Q8 | C2×C4×F5 | C2×C22⋊F5 | D5×C22×C4 | C22⋊F5 | C2×C4×D5 | C22×Dic5 | C22×C20 | C4×D5 | D10 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 4 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C4×C22⋊F5 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 40 | 39 | 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 |
| 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 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 32 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 40 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,1,1,1,1,0,0,0,40,0,0] >;
C4×C22⋊F5 in GAP, Magma, Sage, TeX
C_4\times C_2^2\rtimes F_5
% in TeX
G:=Group("C4xC2^2:F5"); // GroupNames label
G:=SmallGroup(320,1101);
// by ID
G=gap.SmallGroup(320,1101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations