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G = C2×Q8⋊2F5  order 320 = 26·5

Direct product of C2 and Q8⋊2F5

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

Aliases: C2×Q8⋊2F5, C10⋊2C4≀C2, (C2×Q8)⋊5F5, Q8⋊4(C2×F5), D20⋊4(C2×C4), (Q8×C10)⋊5C4, (C2×D20)⋊11C4, Q8⋊2D5⋊6C4, (C4×D5).42D4, D10.7(C2×D4), (C4×F5)⋊6C22, C4.F5⋊3C22, 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), Q8⋊2D5.11C22, C22.51(C22⋊F5), Dic5.46(C22⋊C4), C5⋊2(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×Q8⋊2D5).9C2, (C2×C4×D5).206C22, (C2×C10).59(C22⋊C4), SmallGroup(320,1121)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — C2×Q8⋊2F5
Chief series
C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — C2×C4.F5 — C2×Q8⋊2F5
Lower central
C5 — C10 — C20 — C2×Q8⋊2F5
Upper central
C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×Q8⋊2F5
 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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C42, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, F5, D10, D10, C2×C10, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C5⋊C8, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×F5, C22×D5, C22×D5, C2×C4≀C2, C4.F5, C4.F5, C4×F5, C4×F5, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, Q8⋊2D5, Q8×C10, C22×F5, Q8⋊2F5, C2×C4.F5, C2×C4×F5, C2×Q8⋊2D5, C2×Q8⋊2F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C4≀C2, C2×C22⋊C4, C2×F5, C2×C4≀C2, C22⋊F5, C22×F5, Q8⋊2F5, C2×C22⋊F5, C2×Q8⋊2F5

Smallest permutation representation of C2×Q8⋊2F5
►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⋊F5Q8⋊2F5
kernelC2×Q8⋊2F5Q8⋊2F5C2×C4.F5C2×C4×F5C2×Q8⋊2D5C2×D20Q8⋊2D5Q8×C10C4×D5C2×Dic5C22×D5C10C2×Q8C2×C4Q8C4C22C2
# reps141112422118112222

Matrix representation of C2×Q8⋊2F5 ►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×Q8⋊2F5 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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