Copied to
clipboard

G = C8.25D20order 320 = 26·5

11st non-split extension by C8 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.25D20, C40.46D4, M5(2)⋊3D5, C20.20M4(2), C53(C23.C8), C22.5(C8×D5), (C2×C8).152D10, C8.46(C5⋊D4), (C5×M5(2))⋊7C2, (C2×Dic5).1C8, C20.4C810C2, (C22×D5).1C8, C4.10(C8⋊D5), C10.24(C22⋊C8), (C2×C40).220C22, C2.10(D101C8), C4.42(D10⋊C4), C20.104(C22⋊C4), (C2×C4×D5).1C4, (C2×C52C8).2C4, (C2×C10).18(C2×C8), (C2×C4).137(C4×D5), (C2×C20).225(C2×C4), (C2×C8⋊D5).14C2, SmallGroup(320,72)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C8.25D20
C1C5C10C20C40C2×C40C2×C8⋊D5 — C8.25D20
C5C10C2×C10 — C8.25D20
C1C4C2×C8M5(2)

Generators and relations for C8.25D20
 G = < a,b,c,d | a16=b2=c5=d2=1, bab=a9, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 214 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4 [×2], C4, C22, C22 [×2], C5, C8 [×2], C8, C2×C4, C2×C4 [×3], C23, D5, C10, C10, C16 [×2], C2×C8, C2×C8, M4(2) [×2], C22×C4, Dic5, C20 [×2], D10 [×2], C2×C10, M5(2), M5(2), C2×M4(2), C52C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C23.C8, C52C16, C80, C8⋊D5 [×2], C2×C52C8, C2×C40, C2×C4×D5, C20.4C8, C5×M5(2), C2×C8⋊D5, C8.25D20
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C4×D5, D20, C5⋊D4, C23.C8, C8×D5, C8⋊D5, D10⋊C4, D101C8, C8.25D20

Smallest permutation representation of C8.25D20
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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)
(1 26 43 68 63)(2 27 44 69 64)(3 28 45 70 49)(4 29 46 71 50)(5 30 47 72 51)(6 31 48 73 52)(7 32 33 74 53)(8 17 34 75 54)(9 18 35 76 55)(10 19 36 77 56)(11 20 37 78 57)(12 21 38 79 58)(13 22 39 80 59)(14 23 40 65 60)(15 24 41 66 61)(16 25 42 67 62)
(1 63)(2 56)(3 57)(4 50)(5 51)(6 60)(7 61)(8 54)(9 55)(10 64)(11 49)(12 58)(13 59)(14 52)(15 53)(16 62)(17 75)(18 76)(19 69)(20 70)(21 79)(22 80)(23 73)(24 74)(25 67)(26 68)(27 77)(28 78)(29 71)(30 72)(31 65)(32 66)(33 41)(36 44)(37 45)(40 48)

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

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

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

62 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B8A8B8C8D8E8F10A10B10C10D16A16B16C16D16E16F16G16H20A20B20C20D20E20F40A···40H40I40J40K40L80A···80P
order122244445588888810101010161616161616161620202020202040···404040404080···80
size1122011220222222202022444444202020202222442···244444···4

62 irreducible representations

dim1111111122222222244
type++++++++
imageC1C2C2C2C4C4C8C8D4D5M4(2)D10D20C5⋊D4C4×D5C8⋊D5C8×D5C23.C8C8.25D20
kernelC8.25D20C20.4C8C5×M5(2)C2×C8⋊D5C2×C52C8C2×C4×D5C2×Dic5C22×D5C40M5(2)C20C2×C8C8C8C2×C4C4C22C5C1
# reps1111224422224448828

Matrix representation of C8.25D20 in GL4(𝔽241) generated by

0010
0001
1544300
1988700
,
1000
0100
002400
000240
,
0100
24018900
0001
00240189
,
0100
1000
0001
0010
G:=sub<GL(4,GF(241))| [0,0,154,198,0,0,43,87,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[0,240,0,0,1,189,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C8.25D20 in GAP, Magma, Sage, TeX

C_8._{25}D_{20}
% in TeX

G:=Group("C8.25D20");
// GroupNames label

G:=SmallGroup(320,72);
// by ID

G=gap.SmallGroup(320,72);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,100,570,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽