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, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, C₂×D₄, C₄○D₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₈.C₄, C₈○D₄, C₂×D₈, C₈⋊C₂², C₅⋊₂C₈, C₄₀, Dic₁₀, C₄×D₅, D₂₀, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, D₄.₄D₄, C₈×D₅, C₈⋊D₅, C₄.Dic₅, C₄.Dic₅, D₄⋊D₅, D₄.D₅, C₂×C₄₀, C₅×D₈, C₄○D₂₀, D₄×C₁₀, C₄₀.₆C₄, C₂₀.D₄, D₂₀.₃C₄, D₄.D₁₀, C₁₀×D₈, C₄₀.₂₃D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₄○D₄, D₁₀, C₄⋊D₄, C₅⋊D₄, 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 41 11 71 21 61 31 51)(2 80 12 70 22 60 32 50)(3 79 13 69 23 59 33 49)(4 78 14 68 24 58 34 48)(5 77 15 67 25 57 35 47)(6 76 16 66 26 56 36 46)(7 75 17 65 27 55 37 45)(8 74 18 64 28 54 38 44)(9 73 19 63 29 53 39 43)(10 72 20 62 30 52 40 42)

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

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

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

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

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