C2xES-(2,2):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

Derived series

C₁ — C₂ — 2_-⁽¹⁺⁴⁾ — C₂×2_-⁽¹⁺⁴⁾⋊C₅

Chief series

C₁ — C₂ — 2_-⁽¹⁺⁴⁾ — 2_-⁽¹⁺⁴⁾⋊C₅ — C₂×2_-⁽¹⁺⁴⁾⋊C₅

Lower central

2_-⁽¹⁺⁴⁾ — C₂×2_-⁽¹⁺⁴⁾⋊C₅

Upper central

C₁ — C₂²

Subgroups: 499 in 92 conjugacy classes, 9 normal (7 characteristic)
C₁, C₂, C₂^[×2], C₂^[×2], C₄^[×4], C₂², C₂²^[×4], C₅, C₂×C₄^[×14], D₄^[×8], Q₈^[×8], C₂³, C₁₀^[×3], C₂²×C₄^[×3], C₂×D₄^[×2], C₂×Q₈^[×10], C₄○D₄^[×16], C₂×C₁₀, C₂²×Q₈, C₂×C₄○D₄^[×2], 2_-⁽¹⁺⁴⁾, 2_-⁽¹⁺⁴⁾^[×3], C₂×2_-⁽¹⁺⁴⁾, 2_-⁽¹⁺⁴⁾⋊C₅, C₂×2_-⁽¹⁺⁴⁾⋊C₅

Quotients:
C₁, C₂, C₅, C₁₀, C₂⁴⋊C₅, 2_-⁽¹⁺⁴⁾⋊C₅^[×2], C₂×C₂⁴⋊C₅, C₂×2_-⁽¹⁺⁴⁾⋊C₅

Generators and relations
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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

(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)| (1,4)(2,3)(5,52)(6,53)(7,54)(8,50)(9,51)(10,20)(11,21)(12,22)(13,23)(14,24)(15,31)(16,32)(17,33)(18,34)(19,30)(25,43)(26,44)(27,40)(28,41)(29,42)(35,45)(36,46)(37,47)(38,48)(39,49)(55,60)(56,61)(57,62)(58,63)(59,64), (1,49,2,7)(3,54,4,39)(5,14,47,34)(6,40,48,55)(8,56,45,41)(9,11,46,31)(10,43,30,58)(12,13,32,33)(15,51,21,36)(16,17,22,23)(18,52,24,37)(19,63,20,25)(26,62,64,29)(27,38,60,53)(28,50,61,35)(42,44,57,59), (1,33)(2,13)(3,23)(4,17)(5,41)(6,46)(7,12)(8,34)(9,48)(10,57)(11,40)(14,45)(15,60)(16,39)(18,50)(19,29)(20,62)(21,27)(22,54)(24,35)(25,26)(28,52)(30,42)(31,55)(32,49)(36,53)(37,61)(38,51)(43,44)(47,56)(58,59)(63,64), (1,43,2,58)(3,63,4,25)(5,46,47,9)(6,56,48,41)(7,10,49,30)(8,40,45,55)(11,14,31,34)(12,57,32,42)(13,59,33,44)(15,18,21,24)(16,29,22,62)(17,26,23,64)(19,54,20,39)(27,35,60,50)(28,53,61,38)(36,37,51,52), (1,47,2,5)(3,52,4,37)(6,59,48,44)(7,14,49,34)(8,12,45,32)(9,43,46,58)(10,11,30,31)(13,41,33,56)(15,20,21,19)(16,50,22,35)(17,61,23,28)(18,54,24,39)(25,36,63,51)(26,53,64,38)(27,29,60,62)(40,42,55,57), (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( (1,4)(2,3)(5,52)(6,53)(7,54)(8,50)(9,51)(10,20)(11,21)(12,22)(13,23)(14,24)(15,31)(16,32)(17,33)(18,34)(19,30)(25,43)(26,44)(27,40)(28,41)(29,42)(35,45)(36,46)(37,47)(38,48)(39,49)(55,60)(56,61)(57,62)(58,63)(59,64), (1,49,2,7)(3,54,4,39)(5,14,47,34)(6,40,48,55)(8,56,45,41)(9,11,46,31)(10,43,30,58)(12,13,32,33)(15,51,21,36)(16,17,22,23)(18,52,24,37)(19,63,20,25)(26,62,64,29)(27,38,60,53)(28,50,61,35)(42,44,57,59), (1,33)(2,13)(3,23)(4,17)(5,41)(6,46)(7,12)(8,34)(9,48)(10,57)(11,40)(14,45)(15,60)(16,39)(18,50)(19,29)(20,62)(21,27)(22,54)(24,35)(25,26)(28,52)(30,42)(31,55)(32,49)(36,53)(37,61)(38,51)(43,44)(47,56)(58,59)(63,64), (1,43,2,58)(3,63,4,25)(5,46,47,9)(6,56,48,41)(7,10,49,30)(8,40,45,55)(11,14,31,34)(12,57,32,42)(13,59,33,44)(15,18,21,24)(16,29,22,62)(17,26,23,64)(19,54,20,39)(27,35,60,50)(28,53,61,38)(36,37,51,52), (1,47,2,5)(3,52,4,37)(6,59,48,44)(7,14,49,34)(8,12,45,32)(9,43,46,58)(10,11,30,31)(13,41,33,56)(15,20,21,19)(16,50,22,35)(17,61,23,28)(18,54,24,39)(25,36,63,51)(26,53,64,38)(27,29,60,62)(40,42,55,57), (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([(1,4),(2,3),(5,52),(6,53),(7,54),(8,50),(9,51),(10,20),(11,21),(12,22),(13,23),(14,24),(15,31),(16,32),(17,33),(18,34),(19,30),(25,43),(26,44),(27,40),(28,41),(29,42),(35,45),(36,46),(37,47),(38,48),(39,49),(55,60),(56,61),(57,62),(58,63),(59,64)], [(1,49,2,7),(3,54,4,39),(5,14,47,34),(6,40,48,55),(8,56,45,41),(9,11,46,31),(10,43,30,58),(12,13,32,33),(15,51,21,36),(16,17,22,23),(18,52,24,37),(19,63,20,25),(26,62,64,29),(27,38,60,53),(28,50,61,35),(42,44,57,59)], [(1,33),(2,13),(3,23),(4,17),(5,41),(6,46),(7,12),(8,34),(9,48),(10,57),(11,40),(14,45),(15,60),(16,39),(18,50),(19,29),(20,62),(21,27),(22,54),(24,35),(25,26),(28,52),(30,42),(31,55),(32,49),(36,53),(37,61),(38,51),(43,44),(47,56),(58,59),(63,64)], [(1,43,2,58),(3,63,4,25),(5,46,47,9),(6,56,48,41),(7,10,49,30),(8,40,45,55),(11,14,31,34),(12,57,32,42),(13,59,33,44),(15,18,21,24),(16,29,22,62),(17,26,23,64),(19,54,20,39),(27,35,60,50),(28,53,61,38),(36,37,51,52)], [(1,47,2,5),(3,52,4,37),(6,59,48,44),(7,14,49,34),(8,12,45,32),(9,43,46,58),(10,11,30,31),(13,41,33,56),(15,20,21,19),(16,50,22,35),(17,61,23,28),(18,54,24,39),(25,36,63,51),(26,53,64,38),(27,29,60,62),(40,42,55,57)], [(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 ►G ⊆ GL₅(𝔽₄₁)

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	20	7	29
0	21	39	12	34
0	7	29	39	21
0	12	34	20	2

1	0	0	0	0
0	20	36	13	30
0	5	21	11	28
0	28	11	20	36
0	30	13	5	21

1	0	0	0	0
0	12	34	20	2
0	7	29	39	21
0	20	2	29	7
0	39	21	34	12

37	0	0	0	0
0	1	0	0	0
0	7	29	39	21
0	0	0	0	40
0	21	39	12	34

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] >;

Character table of C₂×2_-⁽¹⁺⁴⁾⋊C₅

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	5A	5B	5C	5D	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	10K	10L
size	1	1	1	1	10	10	10	10	10	10	16	16	16	16	16	16	16	16	16	16	16	16	16	16	16	16
ρ₁	1	1	1	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	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	1	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	ζ₅³	ζ₅⁴	ζ₅²	ζ₅	ζ₅	ζ₅	ζ₅²	ζ₅²	ζ₅³	ζ₅³	ζ₅⁴	ζ₅⁴	linear of order 5
ρ₄	1	1	1	1	1	1	1	1	1	1	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	ζ₅⁴	ζ₅²	ζ₅	ζ₅³	ζ₅³	ζ₅³	ζ₅	ζ₅	ζ₅⁴	ζ₅⁴	ζ₅²	ζ₅²	linear of order 5
ρ₅	1	-1	1	-1	-1	1	1	1	-1	-1	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	ζ₅³	ζ₅⁴	ζ₅²	ζ₅	-ζ₅	-ζ₅	-ζ₅²	-ζ₅²	-ζ₅³	-ζ₅³	-ζ₅⁴	-ζ₅⁴	linear of order 10
ρ₆	1	-1	1	-1	-1	1	1	1	-1	-1	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	ζ₅²	ζ₅	ζ₅³	ζ₅⁴	-ζ₅⁴	-ζ₅⁴	-ζ₅³	-ζ₅³	-ζ₅²	-ζ₅²	-ζ₅	-ζ₅	linear of order 10
ρ₇	1	-1	1	-1	-1	1	1	1	-1	-1	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	ζ₅	ζ₅³	ζ₅⁴	ζ₅²	-ζ₅²	-ζ₅²	-ζ₅⁴	-ζ₅⁴	-ζ₅	-ζ₅	-ζ₅³	-ζ₅³	linear of order 10
ρ₈	1	1	1	1	1	1	1	1	1	1	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	ζ₅	ζ₅³	ζ₅⁴	ζ₅²	ζ₅²	ζ₅²	ζ₅⁴	ζ₅⁴	ζ₅	ζ₅	ζ₅³	ζ₅³	linear of order 5
ρ₉	1	-1	1	-1	-1	1	1	1	-1	-1	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	ζ₅⁴	ζ₅²	ζ₅	ζ₅³	-ζ₅³	-ζ₅³	-ζ₅	-ζ₅	-ζ₅⁴	-ζ₅⁴	-ζ₅²	-ζ₅²	linear of order 10
ρ₁₀	1	1	1	1	1	1	1	1	1	1	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	ζ₅²	ζ₅	ζ₅³	ζ₅⁴	ζ₅⁴	ζ₅⁴	ζ₅³	ζ₅³	ζ₅²	ζ₅²	ζ₅	ζ₅	linear of order 5
ρ₁₁	4	-4	-4	4	0	0	0	0	0	0	-1	-1	-1	-1	1	1	1	1	1	-1	1	-1	1	-1	1	-1	symplectic lifted from 2_-⁽¹⁺⁴⁾⋊C₅, Schur index 2
ρ₁₂	4	4	-4	-4	0	0	0	0	0	0	-1	-1	-1	-1	1	1	1	1	-1	1	-1	1	-1	1	-1	1	symplectic lifted from 2_-⁽¹⁺⁴⁾⋊C₅, Schur index 2
ρ₁₃	4	-4	-4	4	0	0	0	0	0	0	-ζ₅²	-ζ₅	-ζ₅⁴	-ζ₅³	ζ₅²	ζ₅	ζ₅³	ζ₅⁴	ζ₅⁴	-ζ₅⁴	ζ₅³	-ζ₅³	ζ₅²	-ζ₅²	ζ₅	-ζ₅	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₁₄	4	4	-4	-4	0	0	0	0	0	0	-ζ₅²	-ζ₅	-ζ₅⁴	-ζ₅³	ζ₅²	ζ₅	ζ₅³	ζ₅⁴	-ζ₅⁴	ζ₅⁴	-ζ₅³	ζ₅³	-ζ₅²	ζ₅²	-ζ₅	ζ₅	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₁₅	4	4	-4	-4	0	0	0	0	0	0	-ζ₅	-ζ₅³	-ζ₅²	-ζ₅⁴	ζ₅	ζ₅³	ζ₅⁴	ζ₅²	-ζ₅²	ζ₅²	-ζ₅⁴	ζ₅⁴	-ζ₅	ζ₅	-ζ₅³	ζ₅³	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₁₆	4	-4	-4	4	0	0	0	0	0	0	-ζ₅	-ζ₅³	-ζ₅²	-ζ₅⁴	ζ₅	ζ₅³	ζ₅⁴	ζ₅²	ζ₅²	-ζ₅²	ζ₅⁴	-ζ₅⁴	ζ₅	-ζ₅	ζ₅³	-ζ₅³	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₁₇	4	4	-4	-4	0	0	0	0	0	0	-ζ₅³	-ζ₅⁴	-ζ₅	-ζ₅²	ζ₅³	ζ₅⁴	ζ₅²	ζ₅	-ζ₅	ζ₅	-ζ₅²	ζ₅²	-ζ₅³	ζ₅³	-ζ₅⁴	ζ₅⁴	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₁₈	4	-4	-4	4	0	0	0	0	0	0	-ζ₅³	-ζ₅⁴	-ζ₅	-ζ₅²	ζ₅³	ζ₅⁴	ζ₅²	ζ₅	ζ₅	-ζ₅	ζ₅²	-ζ₅²	ζ₅³	-ζ₅³	ζ₅⁴	-ζ₅⁴	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₁₉	4	-4	-4	4	0	0	0	0	0	0	-ζ₅⁴	-ζ₅²	-ζ₅³	-ζ₅	ζ₅⁴	ζ₅²	ζ₅	ζ₅³	ζ₅³	-ζ₅³	ζ₅	-ζ₅	ζ₅⁴	-ζ₅⁴	ζ₅²	-ζ₅²	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₂₀	4	4	-4	-4	0	0	0	0	0	0	-ζ₅⁴	-ζ₅²	-ζ₅³	-ζ₅	ζ₅⁴	ζ₅²	ζ₅	ζ₅³	-ζ₅³	ζ₅³	-ζ₅	ζ₅	-ζ₅⁴	ζ₅⁴	-ζ₅²	ζ₅²	complex lifted from 2_-⁽¹⁺⁴⁾⋊C₅
ρ₂₁	5	-5	5	-5	-1	1	-3	1	-1	3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×C₂⁴⋊C₅
ρ₂₂	5	5	5	5	1	1	-3	1	1	-3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅
ρ₂₃	5	-5	5	-5	3	-3	1	1	-1	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×C₂⁴⋊C₅
ρ₂₄	5	5	5	5	-3	-3	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅
ρ₂₅	5	-5	5	-5	-1	1	1	-3	3	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×C₂⁴⋊C₅
ρ₂₆	5	5	5	5	1	1	1	-3	-3	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊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

G = C₂×2_-⁽¹⁺⁴⁾⋊C₅ order 320 = 2⁶·5

Direct product of C₂ and 2_-⁽¹⁺⁴⁾⋊C₅

G = C2×2- (1+4)⋊C5 order 320 = 26·5

Direct product of C2 and 2- (1+4)⋊C5

G = C₂×2_-⁽¹⁺⁴⁾⋊C₅ order 320 = 2⁶·5

Direct product of C₂ and 2_-⁽¹⁺⁴⁾⋊C₅