Copied to
clipboard

G = C5×C22⋊C8order 160 = 25·5

Direct product of C5 and C22⋊C8

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

Aliases: C5×C22⋊C8, C22⋊C40, C20.65D4, C23.2C20, C10.13M4(2), (C2×C10)⋊3C8, (C2×C8)⋊1C10, (C2×C40)⋊3C2, (C2×C4).3C20, C2.1(C2×C40), C4.16(C5×D4), C10.20(C2×C8), (C2×C20).17C4, (C22×C10).7C4, C22.9(C2×C20), (C22×C20).3C2, (C22×C4).2C10, C2.2(C5×M4(2)), C10.31(C22⋊C4), (C2×C20).135C22, C2.2(C5×C22⋊C4), (C2×C4).31(C2×C10), (C2×C10).58(C2×C4), SmallGroup(160,48)

Series: Derived Chief Lower central Upper central

C1C2 — C5×C22⋊C8
C1C2C4C2×C4C2×C20C2×C40 — C5×C22⋊C8
C1C2 — C5×C22⋊C8
C1C2×C20 — C5×C22⋊C8

Generators and relations for C5×C22⋊C8
 G = < a,b,c,d | a5=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

2C2
2C2
2C22
2C4
2C22
2C10
2C10
2C2×C4
2C8
2C2×C4
2C8
2C2×C10
2C20
2C2×C10
2C40
2C2×C20
2C40
2C2×C20

Smallest permutation representation of C5×C22⋊C8
On 80 points
Generators in S80
(1 59 24 56 16)(2 60 17 49 9)(3 61 18 50 10)(4 62 19 51 11)(5 63 20 52 12)(6 64 21 53 13)(7 57 22 54 14)(8 58 23 55 15)(25 45 73 33 65)(26 46 74 34 66)(27 47 75 35 67)(28 48 76 36 68)(29 41 77 37 69)(30 42 78 38 70)(31 43 79 39 71)(32 44 80 40 72)
(2 32)(4 26)(6 28)(8 30)(9 72)(11 66)(13 68)(15 70)(17 80)(19 74)(21 76)(23 78)(34 51)(36 53)(38 55)(40 49)(42 58)(44 60)(46 62)(48 64)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 72)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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)

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

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

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

C5×C22⋊C8 is a maximal subgroup of
C23.30D20  C53(C23⋊C8)  (C2×Dic5)⋊C8  C22.2D40  Dic5.14M4(2)  Dic5.9M4(2)  C408C4⋊C2  C23.34D20  C23.35D20  C23.10D20  C55(C8×D4)  D107M4(2)  C22⋊C8⋊D5  D104M4(2)  Dic52M4(2)  C52C826D4  D20.31D4  D2013D4  D20.32D4  D2014D4  C23.38D20  C22.D40  C23.13D20  Dic1014D4  C22⋊Dic20  D4×C40

100 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B5C5D8A···8H10A···10L10M···10T20A···20P20Q···20X40A···40AF
order12222244444455558···810···1010···1020···2020···2040···40
size11112211112211112···21···12···21···12···22···2

100 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C4C4C5C8C10C10C20C20C40D4M4(2)C5×D4C5×M4(2)
kernelC5×C22⋊C8C2×C40C22×C20C2×C20C22×C10C22⋊C8C2×C10C2×C8C22×C4C2×C4C23C22C20C10C4C2
# reps12122488488322288

Matrix representation of C5×C22⋊C8 in GL3(𝔽41) generated by

100
0100
0010
,
4000
010
02340
,
100
0400
0040
,
1400
091
0232
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[40,0,0,0,1,23,0,0,40],[1,0,0,0,40,0,0,0,40],[14,0,0,0,9,2,0,1,32] >;

C5×C22⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes C_8
% in TeX

G:=Group("C5xC2^2:C8");
// GroupNames label

G:=SmallGroup(160,48);
// by ID

G=gap.SmallGroup(160,48);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

Export

Subgroup lattice of C5×C22⋊C8 in TeX

׿
×
𝔽