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## G = C5×C42⋊C22order 320 = 26·5

### Direct product of C5 and C42⋊C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C42⋊C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4≀C2 — C5×C42⋊C22
 Lower central C1 — C2 — C4 — C5×C42⋊C22
 Upper central C1 — C20 — C22×C20 — C5×C42⋊C22

Generators and relations for C5×C42⋊C22
G = < a,b,c,d,e | a5=b4=c4=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c, ebe=bc2, cd=dc, ce=ec, de=ed >

Subgroups: 258 in 154 conjugacy classes, 78 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C42⋊C22, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4≀C2, C5×C42⋊C2, C10×M4(2), C10×C4○D4, C5×C42⋊C22
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C42⋊C22, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C42⋊C22

Smallest permutation representation of C5×C42⋊C22
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 36 21 41)(2 37 22 42)(3 38 23 43)(4 39 24 44)(5 40 25 45)(6 66)(7 67)(8 68)(9 69)(10 70)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(26 46 31 51)(27 47 32 52)(28 48 33 53)(29 49 34 54)(30 50 35 55)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 11 76 16)(7 12 77 17)(8 13 78 18)(9 14 79 19)(10 15 80 20)(36 51 41 46)(37 52 42 47)(38 53 43 48)(39 54 44 49)(40 55 45 50)(56 71 61 66)(57 72 62 67)(58 73 63 68)(59 74 64 69)(60 75 65 70)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(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 41)(2 42)(3 43)(4 44)(5 45)(6 56)(7 57)(8 58)(9 59)(10 60)(11 71)(12 72)(13 73)(14 74)(15 75)(16 66)(17 67)(18 68)(19 69)(20 70)(21 36)(22 37)(23 38)(24 39)(25 40)(26 51)(27 52)(28 53)(29 54)(30 55)(31 46)(32 47)(33 48)(34 49)(35 50)(61 76)(62 77)(63 78)(64 79)(65 80)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F ··· 4K 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10X 20A ··· 20H 20I ··· 20T 20U ··· 20AR 40A ··· 40P order 1 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 5 5 5 5 8 8 8 8 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 2 2 4 4 1 1 2 2 2 4 ··· 4 1 1 1 1 4 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C20 C20 C20 D4 D4 C5×D4 C5×D4 C42⋊C22 C5×C42⋊C22 kernel C5×C42⋊C22 C5×C4≀C2 C5×C42⋊C2 C10×M4(2) C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 C42⋊C22 C4≀C2 C42⋊C2 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2×C20 C22×C10 C2×C4 C23 C5 C1 # reps 1 4 1 1 1 2 2 4 4 16 4 4 4 8 8 16 3 1 12 4 2 8

Matrix representation of C5×C42⋊C22 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 1 1 4 36 39 40 2 33 0 0 0 32 0 0 9 0
,
 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
,
 39 36 22 12 0 0 0 1 39 40 2 33 0 1 0 0
,
 40 0 5 36 2 1 39 8 0 0 0 40 0 0 40 0
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,39,0,0,1,40,0,0,4,2,0,9,36,33,32,0],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[39,0,39,0,36,0,40,1,22,0,2,0,12,1,33,0],[40,2,0,0,0,1,0,0,5,39,0,40,36,8,40,0] >;

C5×C42⋊C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes C_2^2
% in TeX

G:=Group("C5xC4^2:C2^2");
// GroupNames label

G:=SmallGroup(320,922);
// by ID

G=gap.SmallGroup(320,922);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1731,7004,3511,172,124]);
// Polycyclic

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

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