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G = C2×2- 1+4⋊C5order 320 = 26·5

Direct product of C2 and 2- 1+4⋊C5

direct product, non-abelian, soluble

Aliases: C2×2- 1+4⋊C5, 2- 1+4⋊C10, (C2×2- 1+4)⋊C5, C22.1(C24⋊C5), C2.2(C2×C24⋊C5), SmallGroup(320,1585)

Series: Derived Chief Lower central Upper central

C1C22- 1+4 — C2×2- 1+4⋊C5
C1C22- 1+42- 1+4⋊C5 — C2×2- 1+4⋊C5
2- 1+4 — C2×2- 1+4⋊C5
C1C22

Generators and relations for C2×2- 1+4⋊C5
 G = < a,b,c,d,e,f | a2=b4=c2=f5=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, fbf-1=b2cde, cd=dc, ce=ec, fcf-1=bc, ede-1=b2d, fdf-1=bcd, fef-1=de >

Subgroups: 499 in 92 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C10, C22×Q8, C2×C4○D4, 2- 1+4, 2- 1+4, C2×2- 1+4, 2- 1+4⋊C5, C2×2- 1+4⋊C5
Quotients: C1, C2, C5, C10, C24⋊C5, 2- 1+4⋊C5, C2×C24⋊C5, C2×2- 1+4⋊C5

Character table of C2×2- 1+4⋊C5

 class 12A2B2C2D2E4A4B4C4D5A5B5C5D10A10B10C10D10E10F10G10H10I10J10K10L
 size 111110101010101016161616161616161616161616161616
ρ111111111111111111111111111    trivial
ρ21-11-1-1111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111111ζ53ζ54ζ5ζ52ζ53ζ54ζ52ζ5ζ5ζ5ζ52ζ52ζ53ζ53ζ54ζ54    linear of order 5
ρ41111111111ζ54ζ52ζ53ζ5ζ54ζ52ζ5ζ53ζ53ζ53ζ5ζ5ζ54ζ54ζ52ζ52    linear of order 5
ρ51-11-1-1111-1-1ζ53ζ54ζ5ζ52ζ53ζ54ζ52ζ555525253535454    linear of order 10
ρ61-11-1-1111-1-1ζ52ζ5ζ54ζ53ζ52ζ5ζ53ζ5454545353525255    linear of order 10
ρ71-11-1-1111-1-1ζ5ζ53ζ52ζ54ζ5ζ53ζ54ζ5252525454555353    linear of order 10
ρ81111111111ζ5ζ53ζ52ζ54ζ5ζ53ζ54ζ52ζ52ζ52ζ54ζ54ζ5ζ5ζ53ζ53    linear of order 5
ρ91-11-1-1111-1-1ζ54ζ52ζ53ζ5ζ54ζ52ζ5ζ5353535554545252    linear of order 10
ρ101111111111ζ52ζ5ζ54ζ53ζ52ζ5ζ53ζ54ζ54ζ54ζ53ζ53ζ52ζ52ζ5ζ5    linear of order 5
ρ114-4-44000000-1-1-1-111111-11-11-11-1    symplectic lifted from 2- 1+4⋊C5, Schur index 2
ρ1244-4-4000000-1-1-1-11111-11-11-11-11    symplectic lifted from 2- 1+4⋊C5, Schur index 2
ρ134-4-440000005255453ζ52ζ5ζ53ζ54ζ5454ζ5353ζ5252ζ55    complex lifted from 2- 1+4⋊C5
ρ1444-4-40000005255453ζ52ζ5ζ53ζ5454ζ5453ζ5352ζ525ζ5    complex lifted from 2- 1+4⋊C5
ρ1544-4-40000005535254ζ5ζ53ζ54ζ5252ζ5254ζ545ζ553ζ53    complex lifted from 2- 1+4⋊C5
ρ164-4-440000005535254ζ5ζ53ζ54ζ52ζ5252ζ5454ζ55ζ5353    complex lifted from 2- 1+4⋊C5
ρ1744-4-40000005354552ζ53ζ54ζ52ζ55ζ552ζ5253ζ5354ζ54    complex lifted from 2- 1+4⋊C5
ρ184-4-440000005354552ζ53ζ54ζ52ζ5ζ55ζ5252ζ5353ζ5454    complex lifted from 2- 1+4⋊C5
ρ194-4-440000005452535ζ54ζ52ζ5ζ53ζ5353ζ55ζ5454ζ5252    complex lifted from 2- 1+4⋊C5
ρ2044-4-40000005452535ζ54ζ52ζ5ζ5353ζ535ζ554ζ5452ζ52    complex lifted from 2- 1+4⋊C5
ρ215-55-5-11-31-130000000000000000    orthogonal lifted from C2×C24⋊C5
ρ22555511-311-30000000000000000    orthogonal lifted from C24⋊C5
ρ235-55-53-311-1-10000000000000000    orthogonal lifted from C2×C24⋊C5
ρ245555-3-311110000000000000000    orthogonal lifted from C24⋊C5
ρ255-55-5-111-33-10000000000000000    orthogonal lifted from C2×C24⋊C5
ρ265555111-3-310000000000000000    orthogonal lifted from C24⋊C5

Smallest permutation representation of C2×2- 1+4⋊C5
On 64 points
Generators in S64
(1 3)(2 4)(5 22)(6 23)(7 24)(8 20)(9 21)(10 44)(11 40)(12 41)(13 42)(14 43)(15 50)(16 51)(17 52)(18 53)(19 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 61)(31 62)(32 63)(33 64)(34 60)(35 46)(36 47)(37 48)(38 49)(39 45)
(1 55 4 53)(2 18 3 25)(5 29 46 17)(6 19 47 26)(7 9 48 45)(8 34 49 44)(10 20 60 38)(11 57 61 50)(12 13 62 63)(14 58 64 51)(15 40 27 30)(16 43 28 33)(21 37 39 24)(22 59 35 52)(23 54 36 56)(31 32 41 42)
(1 63)(2 42)(3 32)(4 13)(5 40)(6 16)(7 34)(8 9)(10 37)(11 22)(12 53)(14 56)(15 29)(17 27)(18 41)(19 33)(20 21)(23 51)(24 60)(25 31)(26 43)(28 47)(30 46)(35 61)(36 58)(38 39)(44 48)(45 49)(50 59)(52 57)(54 64)(55 62)
(1 20 4 38)(2 49 3 8)(5 26 46 19)(6 17 47 29)(7 41 48 31)(9 42 45 32)(10 55 60 53)(11 14 61 64)(12 37 62 24)(13 39 63 21)(15 16 27 28)(18 44 25 34)(22 56 35 54)(23 52 36 59)(30 33 40 43)(50 51 57 58)
(1 58 4 51)(2 16 3 28)(5 7 46 48)(6 32 47 42)(8 27 49 15)(9 17 45 29)(10 11 60 61)(12 56 62 54)(13 23 63 36)(14 55 64 53)(18 43 25 33)(19 41 26 31)(20 57 38 50)(21 52 39 59)(22 24 35 37)(30 44 40 34)
(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)

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

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

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

Matrix representation of C2×2- 1+4⋊C5 in GL5(𝔽41)

400000
040000
004000
000400
000040
,
10000
03952438
05393824
02438236
03824362
,
10000
0220729
021391234
07293921
01234202
,
10000
020361330
05211128
028112036
03013521
,
10000
01234202
07293921
0202297
039213412
,
370000
01000
07293921
000040
021391234

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,39,5,24,38,0,5,39,38,24,0,24,38,2,36,0,38,24,36,2],[1,0,0,0,0,0,2,21,7,12,0,20,39,29,34,0,7,12,39,20,0,29,34,21,2],[1,0,0,0,0,0,20,5,28,30,0,36,21,11,13,0,13,11,20,5,0,30,28,36,21],[1,0,0,0,0,0,12,7,20,39,0,34,29,2,21,0,20,39,29,34,0,2,21,7,12],[37,0,0,0,0,0,1,7,0,21,0,0,29,0,39,0,0,39,0,12,0,0,21,40,34] >;

C2×2- 1+4⋊C5 in GAP, Magma, Sage, TeX

C_2\times 2_-^{1+4}\rtimes C_5
% in TeX

G:=Group("C2xES-(2,2):C5");
// GroupNames label

G:=SmallGroup(320,1585);
// by ID

G=gap.SmallGroup(320,1585);
# by ID

G:=PCGroup([7,-2,-5,-2,2,2,2,-2,849,1270,521,248,1936,718,375,172,3162,1027]);
// Polycyclic

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

Export

Character table of C2×2- 1+4⋊C5 in TeX

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