C8.25D20 - GroupNames

G = C₈.₂₅D₂₀ order 320 = 2⁶·5

11^st 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), C₅:₃(C₂³.C₈), C₂².₅(C₈xD₅), (C₂xC₈).₁₅₂D₁₀, C₈.₄₆(C₅:D₄), (C₅xM₅(2)):₇C₂, (C₂xDic₅).₁C₈, C₂₀.₄C₈:₁₀C₂, (C₂²xD₅).₁C₈, C₄.₁₀(C₈:D₅), C₁₀.₂₄(C₂²:C₈), (C₂xC₄₀).₂₂₀C₂², C₂.₁₀(D₁₀:₁C₈), C₄.₄₂(D₁₀:C₄), C₂₀.₁₀₄(C₂²:C₄), (C₂xC₄xD₅).₁C₄, (C₂xC₅:₂C₈).₂C₄, (C₂xC₁₀).₁₈(C₂xC₈), (C₂xC₄).₁₃₇(C₄xD₅), (C₂xC₂₀).₂₂₅(C₂xC₄), (C₂xC₈:D₅).₁₄C₂, SmallGroup(320,72)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — C₈.₂₅D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄₀ — C₂xC₄₀ — C₂xC₈:D₅ — C₈.₂₅D₂₀

Lower central

C₅ — C₁₀ — C₂xC₁₀ — C₈.₂₅D₂₀

Upper central

C₁ — C₄ — C₂xC₈ — M₅(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, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂xC₄, C₂xC₄, C₂³, D₅, C₁₀, C₁₀, C₁₆, C₂xC₈, C₂xC₈, M₄(2), C₂²xC₄, Dic₅, C₂₀, D₁₀, C₂xC₁₀, M₅(2), M₅(2), C₂xM₄(2), C₅:₂C₈, C₄₀, C₄xD₅, C₂xDic₅, C₂xC₂₀, C₂²xD₅, C₂³.C₈, C₅:₂C₁₆, C₈₀, C₈:D₅, C₂xC₅:₂C₈, C₂xC₄₀, C₂xC₄xD₅, C₂₀.₄C₈, C₅xM₅(2), C₂xC₈:D₅, C₈.₂₅D₂₀
Quotients: C₁, C₂, C₄, C₂², C₈, C₂xC₄, D₄, D₅, C₂²:C₄, C₂xC₈, M₄(2), D₁₀, C₂²:C₈, C₄xD₅, D₂₀, C₅:D₄, C₂³.C₈, C₈xD₅, C₈:D₅, D₁₀:C₄, D₁₀:₁C₈, 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)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)

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

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	5A	5B	8A	8B	8C	8D	8E	8F	10A	10B	10C	10D	16A	16B	16C	16D	16E	16F	16G	16H	20A	20B	20C	20D	20E	20F	40A	···	40H	40I	40J	40K	40L	80A	···	80P
order	1	2	2	2	4	4	4	4	5	5	8	8	8	8	8	8	10	10	10	10	16	16	16	16	16	16	16	16	20	20	20	20	20	20	40	···	40	40	40	40	40	80	···	80
size	1	1	2	20	1	1	2	20	2	2	2	2	2	2	20	20	2	2	4	4	4	4	4	4	20	20	20	20	2	2	2	2	4	4	2	···	2	4	4	4	4	4	···	4

62 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+					+	+		+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	C₈	D₄	D₅	M₄(2)	D₁₀	D₂₀	C₅:D₄	C₄xD₅	C₈:D₅	C₈xD₅	C₂³.C₈	C₈.₂₅D₂₀
kernel	C₈.₂₅D₂₀	C₂₀.₄C₈	C₅xM₅(2)	C₂xC₈:D₅	C₂xC₅:₂C₈	C₂xC₄xD₅	C₂xDic₅	C₂²xD₅	C₄₀	M₅(2)	C₂₀	C₂xC₈	C₈	C₈	C₂xC₄	C₄	C₂²	C₅	C₁
# reps	1	1	1	1	2	2	4	4	2	2	2	2	4	4	4	8	8	2	8

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

0	0	1	0
0	0	0	1
154	43	0	0
198	87	0	0

1	0	0	0
0	1	0	0
0	0	240	0
0	0	0	240

0	1	0	0
240	189	0	0
0	0	0	1
0	0	240	189

0	1	0	0
1	0	0	0
0	0	0	1
0	0	1	0

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

G = C8.25D20 order 320 = 26·5

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

G = C₈.₂₅D₂₀ order 320 = 2⁶·5

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