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G = C5×C22.29C24  order 320 = 26·5

Direct product of C5 and C22.29C24

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

Aliases: C5×C22.29C24, C10.1522+ (1+4), (C2×C20)⋊26D4, C4⋊D4⋊6C10, C4⋊1D4⋊5C10, C42⋊6(C2×C10), C4.16(D4×C10), C22≀C2⋊3C10, (C4×C20)⋊40C22, C4.4D4⋊6C10, C20.323(C2×D4), (C22×D4)⋊7C10, (D4×C10)⋊36C22, C24.16(C2×C10), (Q8×C10)⋊51C22, C22.21(D4×C10), C42⋊C2⋊10C10, (C2×C20).664C23, (C2×C10).355C24, (C22×C20)⋊48C22, C10.190(C22×D4), C23.9(C22×C10), C2.4(C5×2+ (1+4)), (C23×C10).16C22, (C22×C10).91C23, C22.29(C23×C10), (C2×C4)⋊4(C5×D4), (D4×C2×C10)⋊22C2, C4⋊C4⋊14(C2×C10), C2.14(D4×C2×C10), (C2×C4○D4)⋊4C10, (C2×D4)⋊4(C2×C10), (C10×C4○D4)⋊20C2, (C5×C4⋊D4)⋊33C2, (C5×C4⋊1D4)⋊16C2, C22⋊C4⋊4(C2×C10), (C5×C4⋊C4)⋊70C22, (C22×C4)⋊8(C2×C10), (C2×Q8)⋊11(C2×C10), (C5×C22≀C2)⋊13C2, (C2×C10).417(C2×D4), (C5×C4.4D4)⋊26C2, (C5×C42⋊C2)⋊31C2, (C5×C22⋊C4)⋊39C22, (C2×C4).22(C22×C10), SmallGroup(320,1537)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.29C24
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C4⋊1D4 — C5×C22.29C24
Lower central
C1 — C22 — C5×C22.29C24
Upper central
C1 — C2×C10 — C5×C22.29C24

Subgroups: 610 in 334 conjugacy classes, 162 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C5, C2×C4 [×2], C2×C4 [×10], C2×C4 [×4], D4 [×22], Q8 [×2], C23, C23 [×6], C23 [×8], C10, C10 [×2], C10 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×14], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24 [×2], C20 [×4], C20 [×6], C2×C10, C2×C10 [×2], C2×C10 [×28], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, C2×C20 [×2], C2×C20 [×10], C2×C20 [×4], C5×D4 [×22], C5×Q8 [×2], C22×C10, C22×C10 [×6], C22×C10 [×8], C22.29C24, C4×C20 [×2], C5×C22⋊C4 [×10], C5×C4⋊C4 [×2], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×14], D4×C10 [×4], Q8×C10, C5×C4○D4 [×4], C23×C10 [×2], C5×C42⋊C2, C5×C22≀C2 [×4], C5×C4⋊D4 [×4], C5×C4.4D4 [×2], C5×C4⋊1D4 [×2], D4×C2×C10, C10×C4○D4, C5×C22.29C24

Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C22×D4, 2+ (1+4) [×2], C5×D4 [×4], C22×C10 [×15], C22.29C24, D4×C10 [×6], C23×C10, D4×C2×C10, C5×2+ (1+4) [×2], C5×C22.29C24

Generators and relations
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

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

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

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

(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)

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

Matrix representation ►G ⊆ GL6(𝔽41)

100000
010000
0018000
0001800
0000180
0000018
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
910000
2320000
007770
000001
0040343440
000100
,
4000000
0400000
003432320
001771
0000040
000010
,
1320000
0400000
0011402
0004000
000010
0000040
,
100000
010000
0010142
000100
0000400
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,2,0,0,0,0,1,32,0,0,0,0,0,0,7,0,40,0,0,0,7,0,34,1,0,0,7,0,34,0,0,0,0,1,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,1,0,0,0,0,32,7,0,0,0,0,32,7,0,1,0,0,0,1,40,0],[1,0,0,0,0,0,32,40,0,0,0,0,0,0,1,0,0,0,0,0,14,40,0,0,0,0,0,0,1,0,0,0,2,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,14,0,40,0,0,0,2,0,0,40] >;
 

110 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4J5A5B5C5D10A···10L10M···10T10U···10AR20A···20P20Q···20AN
order1222222···244444···4555510···1010···1010···1020···2020···20
size1111224···422224···411111···12···24···42···24···4

110 irreducible representations

dim11111111111111112244
type++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10D4C5×D42+ (1+4)C5×2+ (1+4)
kernelC5×C22.29C24C5×C42⋊C2C5×C22≀C2C5×C4⋊D4C5×C4.4D4C5×C4⋊1D4D4×C2×C10C10×C4○D4C22.29C24C42⋊C2C22≀C2C4⋊D4C4.4D4C4⋊1D4C22×D4C2×C4○D4C2×C20C2×C4C10C2
# reps11442211441616884441628

In GAP, Magma, Sage, TeX 

C_5\times C_2^2._{29}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.29C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1537);
 
// by ID
 
G=gap.SmallGroup(320,1537);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,891,2467]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=f^2=g^2=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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