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G = C10×C22≀C2  order 320 = 26·5

Direct product of C10 and C22≀C2

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

Aliases: C10×C22≀C2, C25⋊2C10, C23⋊5(C5×D4), C24⋊7(C2×C10), (C24×C10)⋊1C2, C22⋊3(D4×C10), (C2×C20)⋊10C23, (C22×D4)⋊3C10, (C22×C10)⋊17D4, (D4×C10)⋊60C22, (C22×C10)⋊3C23, C23⋊1(C22×C10), (C23×C10)⋊2C22, (C2×C10).341C24, (C22×C20)⋊45C22, C10.180(C22×D4), C22.15(C23×C10), C2.4(D4×C2×C10), (D4×C2×C10)⋊18C2, (C2×D4)⋊8(C2×C10), (C2×C10)⋊15(C2×D4), (C2×C22⋊C4)⋊8C10, (C22×C4)⋊5(C2×C10), (C2×C4)⋊1(C22×C10), C22⋊C4⋊10(C2×C10), (C10×C22⋊C4)⋊28C2, (C5×C22⋊C4)⋊64C22, SmallGroup(320,1523)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C10×C22≀C2
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C22≀C2 — C10×C22≀C2
Lower central
C1 — C22 — C10×C22≀C2
Upper central
C1 — C22×C10 — C10×C22≀C2

Subgroups: 1138 in 662 conjugacy classes, 210 normal (12 characteristic)
C1, C2 [×7], C2 [×14], C4 [×6], C22, C22 [×18], C22 [×78], C5, C2×C4 [×6], C2×C4 [×6], D4 [×24], C23, C23 [×20], C23 [×74], C10 [×7], C10 [×14], C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×12], C2×D4 [×12], C24, C24 [×7], C24 [×12], C20 [×6], C2×C10, C2×C10 [×18], C2×C10 [×78], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C2×C20 [×6], C2×C20 [×6], C5×D4 [×24], C22×C10, C22×C10 [×20], C22×C10 [×74], C2×C22≀C2, C5×C22⋊C4 [×12], C22×C20 [×3], D4×C10 [×12], D4×C10 [×12], C23×C10, C23×C10 [×7], C23×C10 [×12], C10×C22⋊C4 [×3], C5×C22≀C2 [×8], D4×C2×C10 [×3], C24×C10, C10×C22≀C2

Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×12], C23 [×15], C10 [×15], C2×D4 [×18], C24, C2×C10 [×35], C22≀C2 [×4], C22×D4 [×3], C5×D4 [×12], C22×C10 [×15], C2×C22≀C2, D4×C10 [×18], C23×C10, C5×C22≀C2 [×4], D4×C2×C10 [×3], C10×C22≀C2

Generators and relations
 G = < a,b,c,d,e,f | a10=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
018000
001800
000230
000023
,
10000
040000
00100
00010
00001
,
10000
01000
00100
00010
000040
,
10000
040000
004000
00010
00001
,
10000
01000
00100
000400
000040
,
400000
004000
040000
00001
00010

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
 

140 conjugacy classes

class 1 2A···2G2H···2S2T2U4A···4F5A5B5C5D10A···10AB10AC···10BX10BY···10CF20A···20X
order12···22···2224···4555510···1010···1010···1020···20
size11···12···2444···411111···12···24···44···4

140 irreducible representations

dim111111111122
type++++++
imageC1C2C2C2C2C5C10C10C10C10D4C5×D4
kernelC10×C22≀C2C10×C22⋊C4C5×C22≀C2D4×C2×C10C24×C10C2×C22≀C2C2×C22⋊C4C22≀C2C22×D4C25C22×C10C23
# reps13831412321241248

In GAP, Magma, Sage, TeX 

C_{10}\times C_2^2\wr C_2
 
% in TeX
 
G:=Group("C10xC2^2wrC2");
 
// GroupNames label
 
G:=SmallGroup(320,1523);
 
// by ID
 
G=gap.SmallGroup(320,1523);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^10=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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