Copied to
clipboard

G = C42.196D10order 320 = 26·5

16th non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.196D10, M4(2).23D10, C4≀C27D5, D4⋊D58C4, C55(C8○D8), Q8⋊D58C4, D4.D58C4, D4.4(C4×D5), C5⋊Q168C4, Q8.4(C4×D5), C4.203(D4×D5), C10.69(C4×D4), C52C8.55D4, D204C47C2, C4○D4.21D10, D20.21(C2×C4), C20.362(C2×D4), D4.Dic52C2, D20.2C49C2, C20.53D46C2, C20.55(C22×C4), (C4×C20).51C22, (C2×C20).264C23, Dic10.22(C2×C4), D4.8D10.2C2, C4○D20.13C22, C4.Dic5.9C22, C22.9(D42D5), C2.23(Dic54D4), (C5×M4(2)).17C22, (C5×C4≀C2)⋊8C2, (C4×C52C8)⋊3C2, C4.20(C2×C4×D5), C52C8.24(C2×C4), (C5×D4).21(C2×C4), (C5×Q8).22(C2×C4), (C5×C4○D4).5C22, (C2×C10).35(C4○D4), (C2×C4).370(C22×D5), (C2×C52C8).230C22, SmallGroup(320,451)

Series: Derived Chief Lower central Upper central

C1C20 — C42.196D10
C1C5C10C20C2×C20C4○D20D4.8D10 — C42.196D10
C5C10C20 — C42.196D10
C1C4C2×C4C4≀C2

Generators and relations for C42.196D10
 G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=ab-1, ad=da, bd=db, dcd-1=b-1c-1 >

Subgroups: 326 in 106 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×6], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, D5, C10, C10 [×2], C42, C2×C8 [×4], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5, C20 [×2], C20 [×3], D10, C2×C10, C2×C10, C4×C8, C4≀C2, C4≀C2, C8.C4, C8○D4 [×2], C4○D8, C52C8 [×4], C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C8○D8, C8×D5, C8⋊D5, C2×C52C8 [×2], C2×C52C8, C4.Dic5, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4×C20, C5×M4(2), C4○D20, C5×C4○D4, C4×C52C8, D204C4, C20.53D4, C5×C4≀C2, D20.2C4, D4.Dic5, D4.8D10, C42.196D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C8○D8, C2×C4×D5, D4×D5, D42D5, Dic54D4, C42.196D10

Smallest permutation representation of C42.196D10
On 80 points
Generators in S80
(1 19)(2 36 20 52)(3 11)(4 38 12 54)(5 13)(6 40 14 56)(7 15)(8 32 16 58)(9 17)(10 34 18 60)(21 73 68 47)(22 69)(23 75 70 49)(24 61)(25 77 62 41)(26 63)(27 79 64 43)(28 65)(29 71 66 45)(30 67)(31 57)(33 59)(35 51)(37 53)(39 55)(42 78)(44 80)(46 72)(48 74)(50 76)
(1 35 19 51)(2 52 20 36)(3 37 11 53)(4 54 12 38)(5 39 13 55)(6 56 14 40)(7 31 15 57)(8 58 16 32)(9 33 17 59)(10 60 18 34)(21 47 68 73)(22 74 69 48)(23 49 70 75)(24 76 61 50)(25 41 62 77)(26 78 63 42)(27 43 64 79)(28 80 65 44)(29 45 66 71)(30 72 67 46)
(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 24 51 50 19 61 35 76)(2 75 36 70 20 49 52 23)(3 22 53 48 11 69 37 74)(4 73 38 68 12 47 54 21)(5 30 55 46 13 67 39 72)(6 71 40 66 14 45 56 29)(7 28 57 44 15 65 31 80)(8 79 32 64 16 43 58 27)(9 26 59 42 17 63 33 78)(10 77 34 62 18 41 60 25)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I5A5B8A8B8C8D8E8F8G···8L8M8N10A10B10C10D10E10F20A20B20C20D20E···20N20O20P40A40B40C40D
order12222444···444558888888···8881010101010102020202020···20202040404040
size112420112···24202244555510···10202022448822224···4888888

56 irreducible representations

dim111111111111222222222444
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D5C4○D4D10D10D10C4×D5C4×D5C8○D8D4×D5D42D5C42.196D10
kernelC42.196D10C4×C52C8D204C4C20.53D4C5×C4≀C2D20.2C4D4.Dic5D4.8D10D4⋊D5D4.D5Q8⋊D5C5⋊Q16C52C8C4≀C2C2×C10C42M4(2)C4○D4D4Q8C5C4C22C1
# reps111111112222222222448228

Matrix representation of C42.196D10 in GL4(𝔽41) generated by

40000
0900
0010
0001
,
9000
03200
0010
0001
,
04000
40000
0066
00351
,
27000
03800
00216
002839
G:=sub<GL(4,GF(41))| [40,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,32,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,6,35,0,0,6,1],[27,0,0,0,0,38,0,0,0,0,2,28,0,0,16,39] >;

C42.196D10 in GAP, Magma, Sage, TeX

C_4^2._{196}D_{10}
% in TeX

G:=Group("C4^2.196D10");
// GroupNames label

G:=SmallGroup(320,451);
// by ID

G=gap.SmallGroup(320,451);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,555,58,136,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations

׿
×
𝔽