C40.23D4 - GroupNames

G = C₄₀.₂₃D₄ order 320 = 2⁶·5

23^rd non-split extension by C₄₀ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄₀.₂₃D₄, D₂₀.₂₅D₄, Dic₁₀.₂₅D₄, (C₁₀×D₈)⋊₂C₂, (C₂×D₈)⋊₁₀D₅, C₄.₆₁(D₄×D₅), (C₂×C₈).₈₇D₁₀, C₄₀.₆C₄⋊₃C₂, (C₂×D₄).₆₇D₁₀, C₂₀.₁₆₉(C₂×D₄), C₂₀.D₄⋊₇C₂, C₅⋊₄(D₄.₄D₄), C₈.₂₇(C₅⋊D₄), D₄.D₁₀⋊₄C₂, D₂₀.₃C₄⋊₁C₂, (C₂×C₄₀).₃₂C₂², C₂.₁₈(C₂₀⋊₂D₄), (C₂×C₂₀).₄₃₇C₂³, C₄○D₂₀.₄₆C₂², (D₄×C₁₀).₈₆C₂², C₁₀.₁₁₁(C₄⋊D₄), C₄.Dic₅.₁₉C₂², C₂².₂₀(D₄⋊₂D₅), C₄.₈₁(C₂×C₅⋊D₄), (C₂×C₄).₁₂₆(C₂²×D₅), (C₂×C₁₀).₁₅₈(C₄○D₄), SmallGroup(320,787)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — C₄₀.₂₃D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄○D₂₀ — D₂₀.₃C₄ — C₄₀.₂₃D₄

Lower central

C₅ — C₁₀ — C₂×C₂₀ — C₄₀.₂₃D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₂×D₈

Generators and relations for C₄₀.₂₃D₄
G = < a,b,c | a⁴⁰=c²=1, b⁴=a²⁰, bab^-1=a^-1, cac=a⁹, cbc=a²⁰b³ >

Subgroups: 398 in 108 conjugacy classes, 37 normal (27 characteristic)
C₁, C₂, C₂^[×4], C₄^[×2], C₄, C₂², C₂²^[×5], C₅, C₈^[×2], C₈^[×3], C₂×C₄, C₂×C₄, D₄^[×6], Q₈, C₂³^[×2], D₅, C₁₀, C₁₀^[×3], C₂×C₈, C₂×C₈, M₄(2)^[×4], D₈^[×4], SD₁₆^[×2], C₂×D₄^[×2], C₄○D₄, Dic₅, C₂₀^[×2], D₁₀, C₂×C₁₀, C₂×C₁₀^[×4], C₄.D₄^[×2], C₈.C₄, C₈○D₄, C₂×D₈, C₈⋊C₂²^[×2], C₅⋊₂C₈^[×3], C₄₀^[×2], Dic₁₀, C₄×D₅, D₂₀, C₅⋊D₄, C₂×C₂₀, C₅×D₄^[×4], C₂²×C₁₀^[×2], D₄.₄D₄, C₈×D₅, C₈⋊D₅, C₄.Dic₅, C₄.Dic₅^[×2], D₄⋊D₅^[×2], D₄.D₅^[×2], C₂×C₄₀, C₅×D₈^[×2], C₄○D₂₀, D₄×C₁₀^[×2], C₄₀.₆C₄, C₂₀.D₄^[×2], D₂₀.₃C₄, D₄.D₁₀^[×2], C₁₀×D₈, C₄₀.₂₃D₄
Quotients: C₁, C₂^[×7], C₂²^[×7], D₄^[×4], C₂³, D₅, C₂×D₄^[×2], C₄○D₄, D₁₀^[×3], C₄⋊D₄, C₅⋊D₄^[×2], C₂²×D₅, D₄.₄D₄, D₄×D₅, D₄⋊₂D₅, C₂×C₅⋊D₄, C₂₀⋊₂D₄, C₄₀.₂₃D₄

Smallest permutation representation of C₄₀.₂₃D₄
►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 51 11 41 21 71 31 61)(2 50 12 80 22 70 32 60)(3 49 13 79 23 69 33 59)(4 48 14 78 24 68 34 58)(5 47 15 77 25 67 35 57)(6 46 16 76 26 66 36 56)(7 45 17 75 27 65 37 55)(8 44 18 74 28 64 38 54)(9 43 19 73 29 63 39 53)(10 42 20 72 30 62 40 52)

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+		+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₅	C₄○D₄	D₁₀	D₁₀	C₅⋊D₄	D₄.₄D₄	D₄×D₅	D₄⋊₂D₅	C₄₀.₂₃D₄
kernel	C₄₀.₂₃D₄	C₄₀.₆C₄	C₂₀.D₄	D₂₀.₃C₄	D₄.D₁₀	C₁₀×D₈	C₄₀	Dic₁₀	D₂₀	C₂×D₈	C₂×C₁₀	C₂×C₈	C₂×D₄	C₈	C₅	C₄	C₂²	C₁
# reps	1	1	2	1	2	1	2	1	1	2	2	2	4	8	2	2	2	8

Matrix representation of C₄₀.₂₃D₄ ►in GL₆(𝔽₄₁)

16	0	0	0	0	0
0	18	0	0	0	0
0	0	0	40	0	0
0	0	1	24	0	0
0	0	0	0	0	40
0	0	0	0	1	24

0	15	0	0	0	0
30	0	0	0	0	0
0	0	0	0	0	40
0	0	0	0	40	0
0	0	17	40	0	0
0	0	1	24	0	0

0	15	0	0	0	0
11	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,40,24,0,0,0,0,0,0,0,1,0,0,0,0,40,24],[0,30,0,0,0,0,15,0,0,0,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,40,0,0,0,0,40,0,0,0],[0,11,0,0,0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₄₀.₂₃D₄ in GAP, Magma, Sage, TeX

C_{40}._{23}D_4

% in TeX

G:=Group("C40.23D4");

// GroupNames label

G:=SmallGroup(320,787);

// by ID

G=gap.SmallGroup(320,787);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,1123,297,136,438,102,12550]);

// Polycyclic

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

// generators/relations

G = C40.23D4 order 320 = 26·5

23rd non-split extension by C40 of D4 acting via D4/C2=C22

G = C₄₀.₂₃D₄ order 320 = 2⁶·5

23^rd non-split extension by C₄₀ of D₄ acting via D₄/C₂=C₂²