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## G = C5×2- 1+4order 160 = 25·5

### Direct product of C5 and 2- 1+4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×2- 1+4
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — C5×C4○D4 — C5×2- 1+4
 Lower central C1 — C2 — C5×2- 1+4
 Upper central C1 — C10 — C5×2- 1+4

Generators and relations for C5×2- 1+4
G = < a,b,c,d,e | a5=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C10, C2×Q8, C4○D4, C20, C2×C10, 2- 1+4, C2×C20, C5×D4, C5×Q8, Q8×C10, C5×C4○D4, C5×2- 1+4
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, 2- 1+4, C22×C10, C23×C10, C5×2- 1+4

Smallest permutation representation of C5×2- 1+4
On 80 points
Generators in S80
(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)
(1 21 11 17)(2 22 12 18)(3 23 13 19)(4 24 14 20)(5 25 15 16)(6 74 80 70)(7 75 76 66)(8 71 77 67)(9 72 78 68)(10 73 79 69)(26 36 35 45)(27 37 31 41)(28 38 32 42)(29 39 33 43)(30 40 34 44)(46 56 55 65)(47 57 51 61)(48 58 52 62)(49 59 53 63)(50 60 54 64)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 74)(7 75)(8 71)(9 72)(10 73)(11 21)(12 22)(13 23)(14 24)(15 25)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 41 11 37)(2 42 12 38)(3 43 13 39)(4 44 14 40)(5 45 15 36)(6 50 80 54)(7 46 76 55)(8 47 77 51)(9 48 78 52)(10 49 79 53)(16 35 25 26)(17 31 21 27)(18 32 22 28)(19 33 23 29)(20 34 24 30)(56 66 65 75)(57 67 61 71)(58 68 62 72)(59 69 63 73)(60 70 64 74)
(1 71 11 67)(2 72 12 68)(3 73 13 69)(4 74 14 70)(5 75 15 66)(6 24 80 20)(7 25 76 16)(8 21 77 17)(9 22 78 18)(10 23 79 19)(26 46 35 55)(27 47 31 51)(28 48 32 52)(29 49 33 53)(30 50 34 54)(36 56 45 65)(37 57 41 61)(38 58 42 62)(39 59 43 63)(40 60 44 64)

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

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

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

C5×2- 1+4 is a maximal subgroup of   2- 1+42D5  2- 1+4.2D5  D20.34C23  D20.35C23  D20.39C23
C5×2- 1+4 is a maximal quotient of   C5×D4×Q8

85 conjugacy classes

 class 1 2A 2B ··· 2F 4A ··· 4J 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10X 20A ··· 20AN order 1 2 2 ··· 2 4 ··· 4 5 5 5 5 10 10 10 10 10 ··· 10 20 ··· 20 size 1 1 2 ··· 2 2 ··· 2 1 1 1 1 1 1 1 1 2 ··· 2 2 ··· 2

85 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + - image C1 C2 C2 C5 C10 C10 2- 1+4 C5×2- 1+4 kernel C5×2- 1+4 Q8×C10 C5×C4○D4 2- 1+4 C2×Q8 C4○D4 C5 C1 # reps 1 5 10 4 20 40 1 4

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

 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 39 0 0 0 1 1 0 0 0 40 40 0 40 0 0 1 1 0
,
 40 0 0 0 0 0 40 39 0 0 0 0 1 0 0 0 0 40 0 1 0 0 1 1 0
,
 40 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 32 0 9 0 0 9 0 0 9
,
 40 0 0 0 0 0 9 0 0 18 0 0 0 32 32 0 0 32 0 9 0 0 0 0 32

G:=sub<GL(5,GF(41))| [10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,1,40,0,0,39,1,40,1,0,0,0,0,1,0,0,0,40,0],[40,0,0,0,0,0,40,0,0,0,0,39,1,40,1,0,0,0,0,1,0,0,0,1,0],[40,0,0,0,0,0,32,0,32,9,0,0,32,0,0,0,0,0,9,0,0,0,0,0,9],[40,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,32,0,0,0,18,32,9,32] >;

C5×2- 1+4 in GAP, Magma, Sage, TeX

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

G:=Group("C5xES-(2,2)");
// GroupNames label

G:=SmallGroup(160,233);
// by ID

G=gap.SmallGroup(160,233);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985,487,764,374,2115]);
// Polycyclic

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

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