Copied to
clipboard

G = C23.18D10order 160 = 25·5

8th non-split extension by C23 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.18D10, (C2×D4).5D5, (C2×C10).7D4, (C2×C4).18D10, C10.47(C2×D4), C23.D58C2, (D4×C10).10C2, C10.29(C4○D4), C10.D414C2, (C2×C20).61C22, (C2×C10).50C23, (C22×Dic5)⋊5C2, C22.4(C5⋊D4), C55(C22.D4), C2.15(D42D5), C22.57(C22×D5), (C22×C10).18C22, (C2×Dic5).17C22, C2.11(C2×C5⋊D4), SmallGroup(160,156)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C23.18D10
C1C5C10C2×C10C2×Dic5C22×Dic5 — C23.18D10
C5C2×C10 — C23.18D10
C1C22C2×D4

Generators and relations for C23.18D10
 G = < a,b,c,d,e | a2=b2=c2=d10=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 208 in 78 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C5, C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×4], C20, C2×C10, C2×C10 [×2], C2×C10 [×5], C22.D4, C2×Dic5 [×4], C2×Dic5 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C10.D4 [×2], C23.D5, C23.D5 [×2], C22×Dic5, D4×C10, C23.18D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C5⋊D4 [×2], C22×D5, D42D5 [×2], C2×C5⋊D4, C23.18D10

Smallest permutation representation of C23.18D10
On 80 points
Generators in S80
(1 21)(2 27)(3 23)(4 29)(5 25)(6 24)(7 30)(8 26)(9 22)(10 28)(11 31)(12 37)(13 33)(14 39)(15 35)(16 36)(17 32)(18 38)(19 34)(20 40)(41 54)(42 72)(43 56)(44 74)(45 58)(46 76)(47 60)(48 78)(49 52)(50 80)(51 66)(53 68)(55 70)(57 62)(59 64)(61 73)(63 75)(65 77)(67 79)(69 71)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)(10 20)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 46)(42 47)(43 48)(44 49)(45 50)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 71)(60 72)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 8)(2 9)(3 10)(4 6)(5 7)(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 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 78)(52 79)(53 80)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)
(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 18 64)(2 45 19 68)(3 49 20 62)(4 43 16 66)(5 47 17 70)(6 61 11 48)(7 65 12 42)(8 69 13 46)(9 63 14 50)(10 67 15 44)(21 71 38 76)(22 58 39 53)(23 79 40 74)(24 56 31 51)(25 77 32 72)(26 54 33 59)(27 75 34 80)(28 52 35 57)(29 73 36 78)(30 60 37 55)

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

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

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

C23.18D10 is a maximal subgroup of
(C2×C20).D4  C23.4D20  C23.5D20  2+ 1+4.2D5  C42.102D10  C42.105D10  C4216D10  C42.118D10  C24.56D10  C24.32D10  C243D10  C24.35D10  C24.36D10  C4⋊C4.178D10  C10.342+ 1+4  C10.352+ 1+4  C10.362+ 1+4  C10.402+ 1+4  C10.732- 1+4  C10.422+ 1+4  C10.432+ 1+4  C10.442+ 1+4  C10.742- 1+4  C10.792- 1+4  C10.802- 1+4  C10.812- 1+4  D5×C22.D4  C10.632+ 1+4  C10.672+ 1+4  C42.137D10  C42.140D10  C4221D10  C42.166D10  C42.168D10  C4228D10  C248D10  C24.42D10  C10.1042- 1+4  C10.1052- 1+4  (C2×C20)⋊15D4  (C2×D12).D5  C30.(C2×D4)  (C2×C10).D12  (S3×C10).D4  C23.22D30
C23.18D10 is a maximal quotient of
C23.42D20  C24.3D10  C24.46D10  C24.7D10  C24.9D10  C23.14D20  C10.97(C4×D4)  (C2×C20).53D4  (C2×C20).55D4  (C2×C10).D8  C4⋊D4.D5  (C2×D4).D10  C22⋊Q8.D5  (C2×C10).Q16  C10.(C4○D8)  C24.18D10  C24.20D10  (C2×D12).D5  C30.(C2×D4)  (C2×C10).D12  (S3×C10).D4  C23.22D30

34 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B10A···10F10G···10N20A20B20C20D
order122222244444445510···1010···1020202020
size11112244101010102020222···24···44444

34 irreducible representations

dim111112222224
type+++++++++-
imageC1C2C2C2C2D4D5C4○D4D10D10C5⋊D4D42D5
kernelC23.18D10C10.D4C23.D5C22×Dic5D4×C10C2×C10C2×D4C10C2×C4C23C22C2
# reps123112242484

Matrix representation of C23.18D10 in GL4(𝔽41) generated by

1000
0100
004022
0001
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
16000
192300
0010
001540
,
183900
192300
0090
001232
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,22,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,19,0,0,0,23,0,0,0,0,1,15,0,0,0,40],[18,19,0,0,39,23,0,0,0,0,9,12,0,0,0,32] >;

C23.18D10 in GAP, Magma, Sage, TeX

C_2^3._{18}D_{10}
% in TeX

G:=Group("C2^3.18D10");
// GroupNames label

G:=SmallGroup(160,156);
// by ID

G=gap.SmallGroup(160,156);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,188,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^10=1,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations

׿
×
𝔽