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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

Aliases: C5×C42⋊C22, C4≀C25C10, C4○D44C20, (C2×Q8)⋊8C20, (D4×C10)⋊34C4, (C2×D4)⋊10C20, C421(C2×C10), (Q8×C10)⋊28C4, D4.7(C2×C20), C4.73(D4×C10), Q8.7(C2×C20), (C4×C20)⋊34C22, (C2×C20).520D4, C20.478(C2×D4), C4.8(C22×C20), C23.12(C5×D4), C42⋊C24C10, C22.13(D4×C10), (C22×C10).30D4, M4(2)⋊10(C2×C10), (C2×M4(2))⋊13C10, (C10×M4(2))⋊31C2, (C2×C20).897C23, C20.212(C22×C4), C20.131(C22⋊C4), (C5×M4(2))⋊39C22, (C22×C20).413C22, (C5×C4≀C2)⋊13C2, (C5×C4○D4)⋊16C4, (C2×C4).25(C5×D4), (C2×C4).23(C2×C20), C4○D4.7(C2×C10), (C2×C4○D4).7C10, (C5×D4).43(C2×C4), C4.16(C5×C22⋊C4), (C5×Q8).46(C2×C4), (C2×C20).369(C2×C4), (C10×C4○D4).21C2, (C2×C10).408(C2×D4), C2.24(C10×C22⋊C4), (C5×C42⋊C2)⋊25C2, C10.153(C2×C22⋊C4), (C2×C4).72(C22×C10), (C22×C4).32(C2×C10), (C5×C4○D4).52C22, C22.22(C5×C22⋊C4), (C2×C10).147(C22⋊C4), SmallGroup(320,922)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C42⋊C22
C1C2C4C2×C4C2×C20C5×M4(2)C5×C4≀C2 — C5×C42⋊C22
C1C2C4 — C5×C42⋊C22
C1C20C22×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 [×5], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C5, C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×5], C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×5], C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C40 [×2], C2×C20 [×6], C2×C20 [×7], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C42⋊C22, C4×C20 [×2], C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2) [×2], C5×M4(2), C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C5×C4≀C2 [×4], C5×C42⋊C2, C10×M4(2), C10×C4○D4, C5×C42⋊C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, C42⋊C22, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], 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 60)(7 56)(8 57)(9 58)(10 59)(11 71)(12 72)(13 73)(14 74)(15 75)(16 61)(17 62)(18 63)(19 64)(20 65)(26 46 34 54)(27 47 35 55)(28 48 31 51)(29 49 32 52)(30 50 33 53)(66 80)(67 76)(68 77)(69 78)(70 79)
(1 27 21 35)(2 28 22 31)(3 29 23 32)(4 30 24 33)(5 26 25 34)(6 80 20 14)(7 76 16 15)(8 77 17 11)(9 78 18 12)(10 79 19 13)(36 47 41 55)(37 48 42 51)(38 49 43 52)(39 50 44 53)(40 46 45 54)(56 67 61 75)(57 68 62 71)(58 69 63 72)(59 70 64 73)(60 66 65 74)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 54)(7 55)(8 51)(9 52)(10 53)(11 42)(12 43)(13 44)(14 45)(15 41)(16 47)(17 48)(18 49)(19 50)(20 46)(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 66)(7 67)(8 68)(9 69)(10 70)(11 57)(12 58)(13 59)(14 60)(15 56)(16 75)(17 71)(18 72)(19 73)(20 74)(21 36)(22 37)(23 38)(24 39)(25 40)(26 54)(27 55)(28 51)(29 52)(30 53)(31 48)(32 49)(33 50)(34 46)(35 47)(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)(6,60)(7,56)(8,57)(9,58)(10,59)(11,71)(12,72)(13,73)(14,74)(15,75)(16,61)(17,62)(18,63)(19,64)(20,65)(26,46,34,54)(27,47,35,55)(28,48,31,51)(29,49,32,52)(30,50,33,53)(66,80)(67,76)(68,77)(69,78)(70,79), (1,27,21,35)(2,28,22,31)(3,29,23,32)(4,30,24,33)(5,26,25,34)(6,80,20,14)(7,76,16,15)(8,77,17,11)(9,78,18,12)(10,79,19,13)(36,47,41,55)(37,48,42,51)(38,49,43,52)(39,50,44,53)(40,46,45,54)(56,67,61,75)(57,68,62,71)(58,69,63,72)(59,70,64,73)(60,66,65,74), (1,56)(2,57)(3,58)(4,59)(5,60)(6,54)(7,55)(8,51)(9,52)(10,53)(11,42)(12,43)(13,44)(14,45)(15,41)(16,47)(17,48)(18,49)(19,50)(20,46)(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)(6,66)(7,67)(8,68)(9,69)(10,70)(11,57)(12,58)(13,59)(14,60)(15,56)(16,75)(17,71)(18,72)(19,73)(20,74)(21,36)(22,37)(23,38)(24,39)(25,40)(26,54)(27,55)(28,51)(29,52)(30,53)(31,48)(32,49)(33,50)(34,46)(35,47)(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)(6,60)(7,56)(8,57)(9,58)(10,59)(11,71)(12,72)(13,73)(14,74)(15,75)(16,61)(17,62)(18,63)(19,64)(20,65)(26,46,34,54)(27,47,35,55)(28,48,31,51)(29,49,32,52)(30,50,33,53)(66,80)(67,76)(68,77)(69,78)(70,79), (1,27,21,35)(2,28,22,31)(3,29,23,32)(4,30,24,33)(5,26,25,34)(6,80,20,14)(7,76,16,15)(8,77,17,11)(9,78,18,12)(10,79,19,13)(36,47,41,55)(37,48,42,51)(38,49,43,52)(39,50,44,53)(40,46,45,54)(56,67,61,75)(57,68,62,71)(58,69,63,72)(59,70,64,73)(60,66,65,74), (1,56)(2,57)(3,58)(4,59)(5,60)(6,54)(7,55)(8,51)(9,52)(10,53)(11,42)(12,43)(13,44)(14,45)(15,41)(16,47)(17,48)(18,49)(19,50)(20,46)(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)(6,66)(7,67)(8,68)(9,69)(10,70)(11,57)(12,58)(13,59)(14,60)(15,56)(16,75)(17,71)(18,72)(19,73)(20,74)(21,36)(22,37)(23,38)(24,39)(25,40)(26,54)(27,55)(28,51)(29,52)(30,53)(31,48)(32,49)(33,50)(34,46)(35,47)(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),(6,60),(7,56),(8,57),(9,58),(10,59),(11,71),(12,72),(13,73),(14,74),(15,75),(16,61),(17,62),(18,63),(19,64),(20,65),(26,46,34,54),(27,47,35,55),(28,48,31,51),(29,49,32,52),(30,50,33,53),(66,80),(67,76),(68,77),(69,78),(70,79)], [(1,27,21,35),(2,28,22,31),(3,29,23,32),(4,30,24,33),(5,26,25,34),(6,80,20,14),(7,76,16,15),(8,77,17,11),(9,78,18,12),(10,79,19,13),(36,47,41,55),(37,48,42,51),(38,49,43,52),(39,50,44,53),(40,46,45,54),(56,67,61,75),(57,68,62,71),(58,69,63,72),(59,70,64,73),(60,66,65,74)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,54),(7,55),(8,51),(9,52),(10,53),(11,42),(12,43),(13,44),(14,45),(15,41),(16,47),(17,48),(18,49),(19,50),(20,46),(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),(6,66),(7,67),(8,68),(9,69),(10,70),(11,57),(12,58),(13,59),(14,60),(15,56),(16,75),(17,71),(18,72),(19,73),(20,74),(21,36),(22,37),(23,38),(24,39),(25,40),(26,54),(27,55),(28,51),(29,52),(30,53),(31,48),(32,49),(33,50),(34,46),(35,47),(61,76),(62,77),(63,78),(64,79),(65,80)])

110 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F···4K5A5B5C5D8A8B8C8D10A10B10C10D10E···10P10Q···10X20A···20H20I···20T20U···20AR40A···40P
order1222222444444···4555588881010101010···1010···1020···2020···2020···2040···40
size1122244112224···41111444411112···24···41···12···24···44···4

110 irreducible representations

dim1111111111111111222244
type+++++++
imageC1C2C2C2C2C4C4C4C5C10C10C10C10C20C20C20D4D4C5×D4C5×D4C42⋊C22C5×C42⋊C22
kernelC5×C42⋊C22C5×C4≀C2C5×C42⋊C2C10×M4(2)C10×C4○D4D4×C10Q8×C10C5×C4○D4C42⋊C22C4≀C2C42⋊C2C2×M4(2)C2×C4○D4C2×D4C2×Q8C4○D4C2×C20C22×C10C2×C4C23C5C1
# reps1411122441644488163112428

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

37000
03700
00370
00037
,
11436
3940233
00032
0090
,
32000
03200
00320
00032
,
39362212
0001
3940233
0100
,
400536
21398
00040
00400
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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