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G = C10×C22≀C2order 320 = 26·5

Direct product of C10 and C22≀C2

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

Aliases: C10×C22≀C2, C252C10, C235(C5×D4), (C24×C10)⋊1C2, C247(C2×C10), C223(D4×C10), (C2×C20)⋊10C23, (C22×D4)⋊3C10, (C22×C10)⋊17D4, (D4×C10)⋊60C22, C231(C22×C10), (C22×C10)⋊3C23, (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), (C10×C22⋊C4)⋊28C2, C22⋊C410(C2×C10), (C5×C22⋊C4)⋊64C22, SmallGroup(320,1523)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C22≀C2
C1C2C22C2×C10C22×C10D4×C10C5×C22≀C2 — C10×C22≀C2
C1C22 — C10×C22≀C2
C1C22×C10 — C10×C22≀C2

Generators and relations for C10×C22≀C2
 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 >

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

Smallest permutation representation of C10×C22≀C2
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 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(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 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
(1 57)(2 58)(3 59)(4 60)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(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 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 61)(50 62)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 61)(10 62)(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 57)(42 58)(43 59)(44 60)(45 51)(46 52)(47 53)(48 54)(49 55)(50 56)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(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 68)(42 69)(43 70)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 51)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(61 77)(62 78)(63 79)(64 80)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)

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

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

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

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

Matrix representation of C10×C22≀C2 in 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] >;

C10×C22≀C2 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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