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G = C5×C22.32C24  order 320 = 26·5

Direct product of C5 and C22.32C24

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

Aliases: C5×C22.32C24, C10.1542+ (1+4), (D4×C20)⋊39C2, (C4×D4)⋊10C10, C4⋊D4⋊8C10, C42⋊7(C2×C10), C22⋊Q8⋊7C10, C22≀C2⋊4C10, (C4×C20)⋊41C22, C4.4D4⋊8C10, C42⋊2C2⋊1C10, (D4×C10)⋊37C22, C24.17(C2×C10), (Q8×C10)⋊28C22, (C2×C20).667C23, (C2×C10).358C24, (C22×C20)⋊49C22, C22.D4⋊4C10, C2.6(C5×2+ (1+4)), C22.32(C23×C10), (C23×C10).17C22, C23.11(C22×C10), (C22×C10).93C23, C4⋊C4⋊15(C2×C10), (C2×D4)⋊5(C2×C10), (C2×Q8)⋊3(C2×C10), (C5×C4⋊D4)⋊35C2, (C5×C4⋊C4)⋊71C22, (C22×C4)⋊9(C2×C10), C2.15(C10×C4○D4), (C5×C22⋊Q8)⋊34C2, (C5×C22≀C2)⋊14C2, C22.4(C5×C4○D4), (C10×C22⋊C4)⋊33C2, (C2×C22⋊C4)⋊13C10, C22⋊C4⋊15(C2×C10), C10.234(C2×C4○D4), (C5×C4.4D4)⋊28C2, (C5×C42⋊2C2)⋊12C2, (C2×C10).52(C4○D4), (C5×C22⋊C4)⋊69C22, (C2×C4).25(C22×C10), (C5×C22.D4)⋊23C2, SmallGroup(320,1540)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.32C24
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C4⋊D4 — C5×C22.32C24
Lower central
C1 — C22 — C5×C22.32C24
Upper central
C1 — C2×C10 — C5×C22.32C24

Subgroups: 434 in 250 conjugacy classes, 146 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], D4 [×9], Q8, C23, C23 [×4], C23 [×4], C10 [×3], C10 [×6], C42 [×2], C22⋊C4 [×14], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C24, C20 [×10], C2×C10, C2×C10 [×2], C2×C10 [×18], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C5×D4 [×9], C5×Q8, C22×C10, C22×C10 [×4], C22×C10 [×4], C22.32C24, C4×C20 [×2], C5×C22⋊C4 [×14], C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C22×C20 [×2], C22×C20 [×2], D4×C10, D4×C10 [×6], Q8×C10, C23×C10, C10×C22⋊C4, D4×C20 [×2], C5×C22≀C2 [×2], C5×C4⋊D4, C5×C4⋊D4 [×2], C5×C22⋊Q8, C5×C22.D4 [×2], C5×C4.4D4 [×2], C5×C42⋊2C2 [×2], C5×C22.32C24

Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×2], C24, C2×C10 [×35], C2×C4○D4, 2+ (1+4) [×2], C22×C10 [×15], C22.32C24, C5×C4○D4 [×2], C23×C10, C10×C4○D4, C5×2+ (1+4) [×2], C5×C22.32C24

Generators and relations
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=g2=1, e2=c, 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 50 30 36)(2 46 26 37)(3 47 27 38)(4 48 28 39)(5 49 29 40)(6 63 76 75)(7 64 77 71)(8 65 78 72)(9 61 79 73)(10 62 80 74)(11 56 16 70)(12 57 17 66)(13 58 18 67)(14 59 19 68)(15 60 20 69)(21 53 33 41)(22 54 34 42)(23 55 35 43)(24 51 31 44)(25 52 32 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), (1,50,30,36)(2,46,26,37)(3,47,27,38)(4,48,28,39)(5,49,29,40)(6,63,76,75)(7,64,77,71)(8,65,78,72)(9,61,79,73)(10,62,80,74)(11,56,16,70)(12,57,17,66)(13,58,18,67)(14,59,19,68)(15,60,20,69)(21,53,33,41)(22,54,34,42)(23,55,35,43)(24,51,31,44)(25,52,32,45), (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), (1,50,30,36)(2,46,26,37)(3,47,27,38)(4,48,28,39)(5,49,29,40)(6,63,76,75)(7,64,77,71)(8,65,78,72)(9,61,79,73)(10,62,80,74)(11,56,16,70)(12,57,17,66)(13,58,18,67)(14,59,19,68)(15,60,20,69)(21,53,33,41)(22,54,34,42)(23,55,35,43)(24,51,31,44)(25,52,32,45), (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)], [(1,50,30,36),(2,46,26,37),(3,47,27,38),(4,48,28,39),(5,49,29,40),(6,63,76,75),(7,64,77,71),(8,65,78,72),(9,61,79,73),(10,62,80,74),(11,56,16,70),(12,57,17,66),(13,58,18,67),(14,59,19,68),(15,60,20,69),(21,53,33,41),(22,54,34,42),(23,55,35,43),(24,51,31,44),(25,52,32,45)], [(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
0040000
0004000
0000400
0000040
,
4000000
210000
000010
000001
001000
000100
,
900000
090000
000100
0040000
0000040
000010
,
40400000
010000
001000
0004000
0000400
000001
,
100000
010000
0040000
0004000
000010
000001

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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

110 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4L5A5B5C5D10A···10L10M···10T10U···10AJ20A···20P20Q···20AV
order122222222244444···4555510···1010···1010···1020···2020···20
size111122444422224···411111···12···24···42···24···4

110 irreducible representations

dim1111111111111111112244
type++++++++++
imageC1C2C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10C10C4○D4C5×C4○D42+ (1+4)C5×2+ (1+4)
kernelC5×C22.32C24C10×C22⋊C4D4×C20C5×C22≀C2C5×C4⋊D4C5×C22⋊Q8C5×C22.D4C5×C4.4D4C5×C42⋊2C2C22.32C24C2×C22⋊C4C4×D4C22≀C2C4⋊D4C22⋊Q8C22.D4C4.4D4C42⋊2C2C2×C10C22C10C2
# reps112231222448812488841628

In GAP, Magma, Sage, TeX 

C_5\times C_2^2._{32}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.32C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1540);
 
// by ID
 
G=gap.SmallGroup(320,1540);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,891,2467]);
 
// 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=c,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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