C40.50D4 - GroupNames

G = C₄₀.₅₀D₄ order 320 = 2⁶·5

50^th non-split extension by C₄₀ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄₀.₅₀D₄, M₄(2).₃₇D₁₀, C₈oD₄:₇D₅, D₄:D₅:₁₀C₄, Q₈:D₅:₁₀C₄, D₄.₈(C₄xD₅), Q₈.₈(C₄xD₅), C₅:₇(C₈.₂₆D₄), D₄.D₅:₁₀C₄, C₄₀:₈C₄:₂₉C₂, C₅:Q₁₆:₁₀C₄, C₄oD₄.₃₅D₁₀, D₂₀.₃₃(C₂xC₄), C₁₀.₁₁₂(C₄xD₄), (C₂xC₈).₁₉₁D₁₀, C₂₀.₄₄₈(C₂xD₄), C₈.₄₇(C₅:D₄), D₂₀:₇C₄:₁₄C₂, D₄:₂Dic₅:₃C₂, C₂₀.₆₅(C₂²xC₄), D₂₀.₃C₄:₁₄C₂, C₂₀.₅₃D₄:₁₄C₂, (C₂xC₄₀).₂₃₇C₂², (C₂xC₂₀).₄₂₅C₂³, Dic₁₀.₃₄(C₂xC₄), D₄.₈D₁₀.₃C₂, C₄oD₂₀.₄₅C₂², C₂².₄(C₄oD₂₀), (C₄xDic₅).₄₇C₂², C₄.Dic₅.₄₅C₂², (C₅xM₄(2)).₄₀C₂², C₄.₃₀(C₂xC₄xD₅), (C₅xC₈oD₄):₇C₂, C₅:₂C₈.₆(C₂xC₄), C₂.₂₇(C₄xC₅:D₄), (C₅xD₄).₂₉(C₂xC₄), C₄.₁₃₉(C₂xC₅:D₄), (C₅xQ₈).₃₀(C₂xC₄), (C₂xC₁₀).₁₀(C₄oD₄), (C₅xC₄oD₄).₄₀C₂², (C₂xC₄).₅₁₅(C₂²xD₅), (C₂xC₅:₂C₈).₁₄₄C₂², SmallGroup(320,772)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₄₀.₅₀D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂xC₂₀ — C₄oD₂₀ — D₄.₈D₁₀ — C₄₀.₅₀D₄

Lower central

C₅ — C₁₀ — C₂₀ — C₄₀.₅₀D₄

Upper central

C₁ — C₄ — C₂xC₈ — C₈oD₄

Generators and relations for C₄₀.₅₀D₄
G = < a,b,c | a⁴⁰=c²=1, b⁴=a²⁰, bab^-1=a²⁹, cac=a⁹, cbc=a²⁰b³ >

Subgroups: 326 in 104 conjugacy classes, 47 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, D₅, C₁₀, C₁₀, C₄², C₂xC₈, C₂xC₈, M₄(2), M₄(2), D₈, SD₁₆, Q₁₆, C₄oD₄, C₄oD₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₈:C₄, C₄wrC₂, C₈.C₄, C₈oD₄, C₈oD₄, C₄oD₈, C₅:₂C₈, C₅:₂C₈, C₄₀, C₄₀, Dic₁₀, C₄xD₅, D₂₀, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₈.₂₆D₄, C₈xD₅, C₈:D₅, C₂xC₅:₂C₈, C₄.Dic₅, C₄xDic₅, D₄:D₅, D₄.D₅, Q₈:D₅, C₅:Q₁₆, C₂xC₄₀, C₂xC₄₀, C₅xM₄(2), C₅xM₄(2), C₄oD₂₀, C₅xC₄oD₄, C₄₀:₈C₄, C₂₀.₅₃D₄, D₂₀:₇C₄, D₄:₂Dic₅, D₂₀.₃C₄, D₄.₈D₁₀, C₅xC₈oD₄, C₄₀.₅₀D₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, D₅, C₂²xC₄, C₂xD₄, C₄oD₄, D₁₀, C₄xD₄, C₄xD₅, C₅:D₄, C₂²xD₅, C₈.₂₆D₄, C₂xC₄xD₅, C₄oD₂₀, C₂xC₅:D₄, C₄xC₅:D₄, C₄₀.₅₀D₄

Smallest permutation representation of C₄₀.₅₀D₄
►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)

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

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	5A	5B	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	10A	10B	10C	···	10H	20A	20B	20C	20D	20E	···	20J	40A	···	40H	40I	···	40T
order	1	2	2	2	2	4	4	4	4	4	4	4	5	5	8	8	8	8	8	8	8	8	8	8	10	10	10	···	10	20	20	20	20	20	···	20	40	···	40	40	···	40
size	1	1	2	4	20	1	1	2	4	20	20	20	2	2	2	2	2	2	4	4	20	20	20	20	2	2	4	···	4	2	2	2	2	4	···	4	2	···	2	4	···	4

62 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+					+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₅	C₄oD₄	D₁₀	D₁₀	D₁₀	C₅:D₄	C₄xD₅	C₄xD₅	C₄oD₂₀	C₈.₂₆D₄	C₄₀.₅₀D₄
kernel	C₄₀.₅₀D₄	C₄₀:₈C₄	C₂₀.₅₃D₄	D₂₀:₇C₄	D₄:₂Dic₅	D₂₀.₃C₄	D₄.₈D₁₀	C₅xC₈oD₄	D₄:D₅	D₄.D₅	Q₈:D₅	C₅:Q₁₆	C₄₀	C₈oD₄	C₂xC₁₀	C₂xC₈	M₄(2)	C₄oD₄	C₈	D₄	Q₈	C₂²	C₅	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	8	4	4	8	2	8

Matrix representation of C₄₀.₅₀D₄ ►in GL₄(F₄₁) generated by

34	34	0	0
7	35	0	0
0	0	7	7
0	0	34	6

13	19	0	0
23	28	0	0
0	0	35	7
0	0	2	6

0	0	35	7
0	0	2	6
13	19	0	0
23	28	0	0

G:=sub<GL(4,GF(41))| [34,7,0,0,34,35,0,0,0,0,7,34,0,0,7,6],[13,23,0,0,19,28,0,0,0,0,35,2,0,0,7,6],[0,0,13,23,0,0,19,28,35,2,0,0,7,6,0,0] >;

C₄₀.₅₀D₄ in GAP, Magma, Sage, TeX

C_{40}._{50}D_4

% in TeX

G:=Group("C40.50D4");

// GroupNames label

G:=SmallGroup(320,772);

// by ID

G=gap.SmallGroup(320,772);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,136,1684,851,102,12550]);

// Polycyclic

G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^29,c*a*c=a^9,c*b*c=a^20*b^3>;

// generators/relations

G = C40.50D4 order 320 = 26·5

50th non-split extension by C40 of D4 acting via D4/C2=C22

G = C₄₀.₅₀D₄ order 320 = 2⁶·5

50^th non-split extension by C₄₀ of D₄ acting via D₄/C₂=C₂²