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G = C2×Q82F5order 320 = 26·5

Direct product of C2 and Q82F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q82F5, C102C4≀C2, (C2×Q8)⋊5F5, Q84(C2×F5), D204(C2×C4), (Q8×C10)⋊5C4, (C2×D20)⋊11C4, Q82D56C4, (C4×D5).42D4, D10.7(C2×D4), (C4×F5)⋊6C22, C4.F53C22, C4.19(C22×F5), C20.19(C22×C4), (C22×D5).69D4, (C4×D5).41C23, C4.18(C22⋊F5), C20.18(C22⋊C4), Dic5.111(C2×D4), (C2×Dic5).261D4, D10.14(C22⋊C4), Q82D5.11C22, C22.51(C22⋊F5), Dic5.46(C22⋊C4), C52(C2×C4≀C2), (C2×C4×F5)⋊3C2, (C5×Q8)⋊4(C2×C4), (C2×C4.F5)⋊3C2, (C2×C4).85(C2×F5), (C2×C20).61(C2×C4), (C4×D5).25(C2×C4), C2.28(C2×C22⋊F5), C10.27(C2×C22⋊C4), (C2×Q82D5).9C2, (C2×C4×D5).206C22, (C2×C10).59(C22⋊C4), SmallGroup(320,1121)

Series: Derived Chief Lower central Upper central

C1C20 — C2×Q82F5
C1C5C10Dic5C4×D5C4.F5C2×C4.F5 — C2×Q82F5
C5C10C20 — C2×Q82F5
C1C22C2×C4C2×Q8

Generators and relations for C2×Q82F5
 G = < a,b,c,d,e | a2=b4=d5=e4=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 746 in 170 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×16], D4 [×7], Q8 [×2], Q8, C23 [×2], D5 [×4], C10, C10 [×2], C42 [×3], C2×C8, M4(2) [×3], C22×C4 [×3], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×6], C2×C10, C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C5⋊C8 [×2], C4×D5 [×4], C4×D5 [×4], D20 [×2], D20 [×5], C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×F5 [×6], C22×D5, C22×D5, C2×C4≀C2, C4.F5 [×2], C4.F5, C4×F5 [×2], C4×F5, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5 [×4], Q82D5 [×2], Q8×C10, C22×F5, Q82F5 [×4], C2×C4.F5, C2×C4×F5, C2×Q82D5, C2×Q82F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4≀C2 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4≀C2, C22⋊F5 [×2], C22×F5, Q82F5 [×2], C2×C22⋊F5, C2×Q82F5

Smallest permutation representation of C2×Q82F5
On 80 points
Generators in S80
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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)(6 8 7 10)(11 13 12 15)(16 18 17 20)(21 38 27 35)(22 40 26 33)(23 37 30 31)(24 39 29 34)(25 36 28 32)(41 43 42 45)(46 48 47 50)(51 53 52 55)(56 58 57 60)(61 78 67 75)(62 80 66 73)(63 77 70 71)(64 79 69 74)(65 76 68 72)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I···4P 5 8A8B8C8D10A10B10C20A···20F
order12222222444444444···45888810101020···20
size1111101020202244555510···104202020204448···8

38 irreducible representations

dim111111112222444448
type++++++++++++++
imageC1C2C2C2C2C4C4C4D4D4D4C4≀C2F5C2×F5C2×F5C22⋊F5C22⋊F5Q82F5
kernelC2×Q82F5Q82F5C2×C4.F5C2×C4×F5C2×Q82D5C2×D20Q82D5Q8×C10C4×D5C2×Dic5C22×D5C10C2×Q8C2×C4Q8C4C22C2
# reps141112422118112222

Matrix representation of C2×Q82F5 in GL8(𝔽41)

10000000
01000000
004000000
000400000
00001000
00000100
00000010
00000001
,
90000000
3732000000
004000000
000400000
00001000
00000100
00000010
00000001
,
4016000000
51000000
00010000
00100000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000040100
000040010
000040001
000040000
,
10000000
209000000
00900000
000320000
00000010
00001000
00000001
00000100

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,37,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,5,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,20,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C2×Q82F5 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_2F_5
% in TeX

G:=Group("C2xQ8:2F5");
// GroupNames label

G:=SmallGroup(320,1121);
// by ID

G=gap.SmallGroup(320,1121);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,136,1684,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^5=e^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations

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