metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊F5⋊2C4, (C22×C4)⋊2F5, (C22×C20)⋊6C4, D10.8(C4⋊C4), C22.2(C4×F5), C22.3(C4⋊F5), C2.2(C23⋊F5), C23.34(C2×F5), (C22×D5).9Q8, (C2×C10).13C42, (C22×Dic5)⋊7C4, (C22×D5).59D4, C5⋊2(C23.9D4), C10.11(C23⋊C4), D10.8(C22⋊C4), C2.3(D10.D4), (C23×D5).82C22, C22.38(C22⋊F5), C2.18(D10.3Q8), C10.17(C2.C42), (C2×C10).16(C4⋊C4), (C2×C22⋊F5).2C2, (C2×D10⋊C4).2C2, (C22×C10).46(C2×C4), (C22×D5).38(C2×C4), (C2×C10).30(C22⋊C4), SmallGroup(320,255)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊F5⋊C4
G = < a,b,c,d,e | a2=b2=c5=d4=e4=1, dad-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c3, ce=ec, ede-1=abd >
Subgroups: 786 in 142 conjugacy classes, 38 normal (24 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C2×Dic5, C2×C20, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C23.9D4, D10⋊C4, C22⋊F5, C22⋊F5, C22×Dic5, C22×C20, C22×F5, C23×D5, C2×D10⋊C4, C2×C22⋊F5, C22⋊F5⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C23⋊C4, C2×F5, C23.9D4, C4×F5, C4⋊F5, C22⋊F5, D10.D4, D10.3Q8, C23⋊F5, C22⋊F5⋊C4
►On 80 points
Generators in S80
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(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 64 34 59)(2 61 33 57)(3 63 32 60)(4 65 31 58)(5 62 35 56)(6 68 37 55)(7 70 36 53)(8 67 40 51)(9 69 39 54)(10 66 38 52)(11 78 22 45)(12 80 21 43)(13 77 25 41)(14 79 24 44)(15 76 23 42)(16 73 27 50)(17 75 26 48)(18 72 30 46)(19 74 29 49)(20 71 28 47)
(1 64 24 44)(2 65 25 45)(3 61 21 41)(4 62 22 42)(5 63 23 43)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)
G:=sub<Sym(80)| (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,64,34,59)(2,61,33,57)(3,63,32,60)(4,65,31,58)(5,62,35,56)(6,68,37,55)(7,70,36,53)(8,67,40,51)(9,69,39,54)(10,66,38,52)(11,78,22,45)(12,80,21,43)(13,77,25,41)(14,79,24,44)(15,76,23,42)(16,73,27,50)(17,75,26,48)(18,72,30,46)(19,74,29,49)(20,71,28,47), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)>;
G:=Group( (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,64,34,59)(2,61,33,57)(3,63,32,60)(4,65,31,58)(5,62,35,56)(6,68,37,55)(7,70,36,53)(8,67,40,51)(9,69,39,54)(10,66,38,52)(11,78,22,45)(12,80,21,43)(13,77,25,41)(14,79,24,44)(15,76,23,42)(16,73,27,50)(17,75,26,48)(18,72,30,46)(19,74,29,49)(20,71,28,47), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60) );
G=PermutationGroup([[(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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)], [(1,64,34,59),(2,61,33,57),(3,63,32,60),(4,65,31,58),(5,62,35,56),(6,68,37,55),(7,70,36,53),(8,67,40,51),(9,69,39,54),(10,66,38,52),(11,78,22,45),(12,80,21,43),(13,77,25,41),(14,79,24,44),(15,76,23,42),(16,73,27,50),(17,75,26,48),(18,72,30,46),(19,74,29,49),(20,71,28,47)], [(1,64,24,44),(2,65,25,45),(3,61,21,41),(4,62,22,42),(5,63,23,43),(6,66,26,46),(7,67,27,47),(8,68,28,48),(9,69,29,49),(10,70,30,50),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4L | 5 | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 20 | ··· | 20 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | F5 | C23⋊C4 | C2×F5 | C4×F5 | C4⋊F5 | C22⋊F5 | D10.D4 | C23⋊F5 |
| kernel | C22⋊F5⋊C4 | C2×D10⋊C4 | C2×C22⋊F5 | C22⋊F5 | C22×Dic5 | C22×C20 | C22×D5 | C22×D5 | C22×C4 | C10 | C23 | C22 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 3 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C22⋊F5⋊C4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 3 | 3 |
| 0 | 0 | 38 | 19 | 38 | 0 |
| 0 | 0 | 0 | 38 | 19 | 38 |
| 0 | 0 | 3 | 3 | 0 | 22 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 16 | 37 | 0 | 0 | 0 | 0 |
| 13 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 7 | 40 |
| 0 | 0 | 1 | 34 | 35 | 38 |
| 0 | 0 | 1 | 4 | 7 | 8 |
| 0 | 0 | 3 | 4 | 37 | 38 |
| 21 | 5 | 0 | 0 | 0 | 0 |
| 10 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 33 | 40 | 34 |
| 0 | 0 | 7 | 3 | 40 | 6 |
| 0 | 0 | 35 | 1 | 38 | 34 |
| 0 | 0 | 7 | 1 | 8 | 4 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[16,13,0,0,0,0,37,25,0,0,0,0,0,0,3,1,1,3,0,0,6,34,4,4,0,0,7,35,7,37,0,0,40,38,8,38],[21,10,0,0,0,0,5,20,0,0,0,0,0,0,37,7,35,7,0,0,33,3,1,1,0,0,40,40,38,8,0,0,34,6,34,4] >;
C22⋊F5⋊C4 in GAP, Magma, Sage, TeX
C_2^2\rtimes F_5\rtimes C_4
% in TeX
G:=Group("C2^2:F5:C4"); // GroupNames label
G:=SmallGroup(320,255);
// by ID
G=gap.SmallGroup(320,255);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^4=e^4=1,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^3,c*e=e*c,e*d*e^-1=a*b*d>;
// generators/relations