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(C₈×D₅), (C₂×C₈).₁₅₂D₁₀, C₈.₄₆(C₅⋊D₄), (C₅×M₅(2))⋊₇C₂, (C₂×Dic₅).₁C₈, C₂₀.₄C₈⋊₁₀C₂, (C₂²×D₅).₁C₈, C₄.₁₀(C₈⋊D₅), C₁₀.₂₄(C₂²⋊C₈), (C₂×C₄₀).₂₂₀C₂², C₂.₁₀(D₁₀⋊₁C₈), C₄.₄₂(D₁₀⋊C₄), C₂₀.₁₀₄(C₂²⋊C₄), (C₂×C₄×D₅).₁C₄, (C₂×C₅⋊₂C₈).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

Derived series

C₁ — C₂×C₁₀ — C₈.₂₅D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄₀ — C₂×C₄₀ — C₂×C₈⋊D₅ — C₈.₂₅D₂₀

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₈.₂₅D₂₀

Upper central

C₁ — C₄ — C₂×C₈ — 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^[×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), C₅⋊₂C₈, C₄₀^[×2], C₄×D₅^[×2], C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₂³.C₈, C₅⋊₂C₁₆, C₈₀, C₈⋊D₅^[×2], C₂×C₅⋊₂C₈, 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, 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)(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	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	C₄×D₅	C₈⋊D₅	C₈×D₅	C₂³.C₈	C₈.₂₅D₂₀
kernel	C₈.₂₅D₂₀	C₂₀.₄C₈	C₅×M₅(2)	C₂×C₈⋊D₅	C₂×C₅⋊₂C₈	C₂×C₄×D₅	C₂×Dic₅	C₂²×D₅	C₄₀	M₅(2)	C₂₀	C₂×C₈	C₈	C₈	C₂×C₄	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