Copied to
clipboard

G = C2×D5×M4(2)  order 320 = 26·5

Direct product of C2, D5 and M4(2)

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D5×M4(2), C407C23, C20.68C24, (C2×C8)⋊29D10, C87(C22×D5), (C2×C40)⋊23C22, C52C812C23, (C8×D5)⋊21C22, C105(C2×M4(2)), C4.67(C23×D5), C23.58(C4×D5), C55(C22×M4(2)), C8⋊D517C22, (C10×M4(2))⋊8C2, C10.52(C23×C4), (C23×D5).12C4, (C4×D5).95C23, (C2×C20).881C23, C20.150(C22×C4), D10.55(C22×C4), (C22×C4).373D10, C4.Dic525C22, (C5×M4(2))⋊24C22, (C22×Dic5).24C4, Dic5.57(C22×C4), (C22×C20).263C22, (D5×C2×C8)⋊28C2, (C2×C4×D5).14C4, C4.122(C2×C4×D5), (C2×C8⋊D5)⋊26C2, (D5×C22×C4).8C2, C2.32(D5×C22×C4), C22.76(C2×C4×D5), (C4×D5).78(C2×C4), (C2×C4).162(C4×D5), (C2×C20).303(C2×C4), (C2×C52C8)⋊47C22, (C2×C4.Dic5)⋊24C2, (C2×C4×D5).323C22, (C2×C4).604(C22×D5), (C2×C10).125(C22×C4), (C22×C10).145(C2×C4), (C2×Dic5).160(C2×C4), (C22×D5).112(C2×C4), SmallGroup(320,1415)

Series: Derived Chief Lower central Upper central

C1C10 — C2×D5×M4(2)
C1C5C10C20C4×D5C2×C4×D5D5×C22×C4 — C2×D5×M4(2)
C5C10 — C2×D5×M4(2)
C1C2×C4C2×M4(2)

Generators and relations for C2×D5×M4(2)
 G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 862 in 298 conjugacy classes, 159 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×20], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×13], C24, Dic5 [×4], C20 [×2], C20 [×2], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8 [×2], C2×M4(2), C2×M4(2) [×11], C23×C4, C52C8 [×4], C40 [×4], C4×D5 [×16], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C22×M4(2), C8×D5 [×8], C8⋊D5 [×8], C2×C52C8 [×2], C4.Dic5 [×4], C2×C40 [×2], C5×M4(2) [×4], C2×C4×D5 [×4], C2×C4×D5 [×8], C22×Dic5, C22×C20, C23×D5, D5×C2×C8 [×2], C2×C8⋊D5 [×2], D5×M4(2) [×8], C2×C4.Dic5, C10×M4(2), D5×C22×C4, C2×D5×M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, M4(2) [×4], C22×C4 [×14], C24, D10 [×7], C2×M4(2) [×6], C23×C4, C4×D5 [×4], C22×D5 [×7], C22×M4(2), C2×C4×D5 [×6], C23×D5, D5×M4(2) [×2], D5×C22×C4, C2×D5×M4(2)

Smallest permutation representation of C2×D5×M4(2)
On 80 points
Generators in S80
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(1 66 12 28 60)(2 67 13 29 61)(3 68 14 30 62)(4 69 15 31 63)(5 70 16 32 64)(6 71 9 25 57)(7 72 10 26 58)(8 65 11 27 59)(17 36 79 52 43)(18 37 80 53 44)(19 38 73 54 45)(20 39 74 55 46)(21 40 75 56 47)(22 33 76 49 48)(23 34 77 50 41)(24 35 78 51 42)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 71)(26 72)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H8I···8P10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order122222222222444444444444558···88···810···101010101020···202020202040···40
size1111225555101011112255551010222···210···102···244442···244444···4

80 irreducible representations

dim111111111122222224
type+++++++++++
imageC1C2C2C2C2C2C2C4C4C4D5M4(2)D10D10D10C4×D5C4×D5D5×M4(2)
kernelC2×D5×M4(2)D5×C2×C8C2×C8⋊D5D5×M4(2)C2×C4.Dic5C10×M4(2)D5×C22×C4C2×C4×D5C22×Dic5C23×D5C2×M4(2)D10C2×C8M4(2)C22×C4C2×C4C23C2
# reps12281111222284821248

Matrix representation of C2×D5×M4(2) in GL5(𝔽41)

400000
01000
00100
00010
00001
,
10000
034100
040000
00010
00001
,
400000
00100
01000
00010
00001
,
400000
032000
003200
0003236
000189
,
400000
01000
00100
00011
000040

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,18,0,0,0,36,9],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,40] >;

C2×D5×M4(2) in GAP, Magma, Sage, TeX

C_2\times D_5\times M_4(2)
% in TeX

G:=Group("C2xD5xM4(2)");
// GroupNames label

G:=SmallGroup(320,1415);
// by ID

G=gap.SmallGroup(320,1415);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,80,102,12550]);
// Polycyclic

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

׿
×
𝔽