metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×Q8)⋊2F5, (Q8×C10)⋊2C4, (C4×Dic5)⋊6C4, C5⋊2(C42⋊3C4), C2.9(C23⋊F5), (C22×D5).13D4, C10.18(C23⋊C4), D10.D4.2C2, C20.23D4.3C2, (C2×D20).41C22, C22.21(C22⋊F5), (C2×C4).3(C2×F5), (C2×C20).16(C2×C4), (C2×C10).41(C22⋊C4), SmallGroup(320,266)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×Q8)⋊F5
G = < a,b,c,d,e | a2=b4=d5=e4=1, c2=b2, ebe-1=ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d3 >
Subgroups: 458 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, F5, D10, C2×C10, C23⋊C4, C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×F5, C22×D5, C42⋊3C4, C4×Dic5, D10⋊C4, C22⋊F5, C2×D20, Q8×C10, D10.D4, C20.23D4, (C2×Q8)⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42⋊3C4, C22⋊F5, C23⋊F5, (C2×Q8)⋊F5
Character table of (C2×Q8)⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 20E | 20F | |
| size | 1 | 1 | 2 | 20 | 20 | 4 | 8 | 20 | 20 | 40 | 40 | 40 | 40 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ12 | 4 | 4 | 4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ13 | 4 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -√5 | √5 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ15 | 4 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | √5 | -√5 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊3C4 |
| ρ17 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | √5 | -√5 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | complex lifted from C23⋊F5 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊3C4 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | √5 | -√5 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | complex lifted from C23⋊F5 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -√5 | √5 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | complex lifted from C23⋊F5 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -√5 | √5 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | complex lifted from C23⋊F5 |
| ρ22 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2√5 | 2√5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2√5 | -2√5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 80 points
Generators in S80
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 71 46 76)(42 72 47 77)(43 73 48 78)(44 74 49 79)(45 75 50 80)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)
(1 56 6 51)(2 57 7 52)(3 58 8 53)(4 59 9 54)(5 60 10 55)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(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)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 36 26 31)(22 38 30 34)(23 40 29 32)(24 37 28 35)(25 39 27 33)(41 61 51 76)(42 63 55 79)(43 65 54 77)(44 62 53 80)(45 64 52 78)(46 66 56 71)(47 68 60 74)(48 70 59 72)(49 67 58 75)(50 69 57 73)
G:=sub<Sym(80)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,36,6,31)(2,37,7,32)(3,38,8,33)(4,39,9,34)(5,40,10,35)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25)(41,71,46,76)(42,72,47,77)(43,73,48,78)(44,74,49,79)(45,75,50,80)(51,61,56,66)(52,62,57,67)(53,63,58,68)(54,64,59,69)(55,65,60,70), (1,56,6,51)(2,57,7,52)(3,58,8,53)(4,59,9,54)(5,60,10,55)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,76,26,71)(22,77,27,72)(23,78,28,73)(24,79,29,74)(25,80,30,75)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,36,26,31)(22,38,30,34)(23,40,29,32)(24,37,28,35)(25,39,27,33)(41,61,51,76)(42,63,55,79)(43,65,54,77)(44,62,53,80)(45,64,52,78)(46,66,56,71)(47,68,60,74)(48,70,59,72)(49,67,58,75)(50,69,57,73)>;
G:=Group( (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,36,6,31)(2,37,7,32)(3,38,8,33)(4,39,9,34)(5,40,10,35)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25)(41,71,46,76)(42,72,47,77)(43,73,48,78)(44,74,49,79)(45,75,50,80)(51,61,56,66)(52,62,57,67)(53,63,58,68)(54,64,59,69)(55,65,60,70), (1,56,6,51)(2,57,7,52)(3,58,8,53)(4,59,9,54)(5,60,10,55)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,76,26,71)(22,77,27,72)(23,78,28,73)(24,79,29,74)(25,80,30,75)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,36,26,31)(22,38,30,34)(23,40,29,32)(24,37,28,35)(25,39,27,33)(41,61,51,76)(42,63,55,79)(43,65,54,77)(44,62,53,80)(45,64,52,78)(46,66,56,71)(47,68,60,74)(48,70,59,72)(49,67,58,75)(50,69,57,73) );
G=PermutationGroup([[(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,36,6,31),(2,37,7,32),(3,38,8,33),(4,39,9,34),(5,40,10,35),(11,26,16,21),(12,27,17,22),(13,28,18,23),(14,29,19,24),(15,30,20,25),(41,71,46,76),(42,72,47,77),(43,73,48,78),(44,74,49,79),(45,75,50,80),(51,61,56,66),(52,62,57,67),(53,63,58,68),(54,64,59,69),(55,65,60,70)], [(1,56,6,51),(2,57,7,52),(3,58,8,53),(4,59,9,54),(5,60,10,55),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,76,26,71),(22,77,27,72),(23,78,28,73),(24,79,29,74),(25,80,30,75),(31,66,36,61),(32,67,37,62),(33,68,38,63),(34,69,39,64),(35,70,40,65)], [(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)], [(2,3,5,4),(7,8,10,9),(11,16),(12,18,15,19),(13,20,14,17),(21,36,26,31),(22,38,30,34),(23,40,29,32),(24,37,28,35),(25,39,27,33),(41,61,51,76),(42,63,55,79),(43,65,54,77),(44,62,53,80),(45,64,52,78),(46,66,56,71),(47,68,60,74),(48,70,59,72),(49,67,58,75),(50,69,57,73)]])
Matrix representation of (C2×Q8)⋊F5 ►in GL8(𝔽41)
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 7 | 0 | 5 | 0 | 0 | 0 | 0 |
| 7 | 35 | 5 | 0 | 0 | 0 | 0 | 0 |
| 25 | 32 | 6 | 34 | 0 | 0 | 0 | 0 |
| 32 | 25 | 34 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 38 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 19 | 38 | 3 |
| 0 | 0 | 0 | 0 | 3 | 38 | 19 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 38 | 22 |
| 28 | 22 | 39 | 2 | 0 | 0 | 0 | 0 |
| 22 | 28 | 2 | 39 | 0 | 0 | 0 | 0 |
| 10 | 11 | 13 | 19 | 0 | 0 | 0 | 0 |
| 11 | 10 | 19 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 32 | 31 | 32 |
| 0 | 0 | 0 | 0 | 28 | 5 | 22 | 22 |
| 0 | 0 | 0 | 0 | 19 | 19 | 36 | 13 |
| 0 | 0 | 0 | 0 | 9 | 10 | 9 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 33 | 0 | 1 | 0 | 0 | 0 | 0 |
| 8 | 19 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[35,7,25,32,0,0,0,0,7,35,32,25,0,0,0,0,0,5,6,34,0,0,0,0,5,0,34,6,0,0,0,0,0,0,0,0,22,0,3,3,0,0,0,0,38,19,38,0,0,0,0,0,0,38,19,38,0,0,0,0,3,3,0,22],[28,22,10,11,0,0,0,0,22,28,11,10,0,0,0,0,39,2,13,19,0,0,0,0,2,39,19,13,0,0,0,0,0,0,0,0,14,28,19,9,0,0,0,0,32,5,19,10,0,0,0,0,31,22,36,9,0,0,0,0,32,22,13,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[1,0,22,8,0,0,0,0,0,40,33,19,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
(C2×Q8)⋊F5 in GAP, Magma, Sage, TeX
(C_2\times Q_8)\rtimes F_5
% in TeX
G:=Group("(C2xQ8):F5"); // GroupNames label
G:=SmallGroup(320,266);
// by ID
G=gap.SmallGroup(320,266);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,297,136,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^5=e^4=1,c^2=b^2,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations
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