D40:14C4 - GroupNames

G = D₄₀⋊₁₄C₄ order 320 = 2⁶·5

8^th semidirect product of D₄₀ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄₀⋊₁₄C₄, C₄₀.₃₈D₄, C₈.₂₂D₂₀, C₂₀.₃SD₁₆, Dic₂₀⋊₁₄C₄, C₈.₁(C₄×D₅), C₄.Q₈⋊₁D₅, C₅⋊₃(D₈⋊₂C₄), C₄₀.₆₂(C₂×C₄), (C₂×C₁₀).₃₀D₈, (C₂×C₈).₄₁D₁₀, (C₂×C₂₀).₈₈D₄, C₄.₇(Q₈⋊D₅), C₂₀.₄C₈⋊₄C₂, D₄₀⋊₇C₂.₆C₂, (C₂×C₄₀).₄₇C₂², C₂².₈(D₄⋊D₅), C₄.₁(D₁₀⋊C₄), C₂.₆(D₂₀⋊₆C₄), C₂₀.₄₈(C₂²⋊C₄), C₁₀.₁₉(D₄⋊C₄), (C₅×C₄.Q₈)⋊₁C₂, (C₂×C₄).₁₆(C₅⋊D₄), SmallGroup(320,46)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₀ — D₄₀⋊₁₄C₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₂×C₄₀ — D₄₀⋊₇C₂ — D₄₀⋊₁₄C₄

Lower central

C₅ — C₁₀ — C₂₀ — C₄₀ — D₄₀⋊₁₄C₄

Upper central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₄.Q₈

Generators and relations for D₄₀⋊₁₄C₄
G = < a,b,c | a⁴⁰=b²=c⁴=1, bab=a^-1, cac^-1=a¹¹, cbc^-1=a¹⁵b >

Subgroups: 286 in 58 conjugacy classes, 25 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, D₅, C₁₀, C₁₀, C₁₆, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₄.Q₈, M₅(2), C₄○D₈, C₄₀, Dic₁₀, C₄×D₅, D₂₀, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, D₈⋊₂C₄, C₅⋊₂C₁₆, C₄₀⋊C₂, D₄₀, Dic₂₀, C₅×C₄⋊C₄, C₂×C₄₀, C₄○D₂₀, C₂₀.₄C₈, C₅×C₄.Q₈, D₄₀⋊₇C₂, D₄₀⋊₁₄C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₈, SD₁₆, D₁₀, D₄⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, D₈⋊₂C₄, D₁₀⋊C₄, D₄⋊D₅, Q₈⋊D₅, D₂₀⋊₆C₄, D₄₀⋊₁₄C₄

Smallest permutation representation of D₄₀⋊₁₄C₄
►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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	5A	5B	8A	8B	8C	10A	···	10F	16A	16B	16C	16D	20A	20B	20C	20D	20E	···	20L	40A	···	40H
order	1	2	2	2	4	4	4	4	4	5	5	8	8	8	10	···	10	16	16	16	16	20	20	20	20	20	···	20	40	···	40
size	1	1	2	40	2	2	8	8	40	2	2	2	2	4	2	···	2	20	20	20	20	4	4	4	4	8	···	8	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+			+	+	+		+	+		+			+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	SD₁₆	D₈	D₁₀	C₄×D₅	D₂₀	C₅⋊D₄	D₈⋊₂C₄	Q₈⋊D₅	D₄⋊D₅	D₄₀⋊₁₄C₄
kernel	D₄₀⋊₁₄C₄	C₂₀.₄C₈	C₅×C₄.Q₈	D₄₀⋊₇C₂	D₄₀	Dic₂₀	C₄₀	C₂×C₂₀	C₄.Q₈	C₂₀	C₂×C₁₀	C₂×C₈	C₈	C₈	C₂×C₄	C₅	C₄	C₂²	C₁
# reps	1	1	1	1	2	2	1	1	2	2	2	2	4	4	4	2	2	2	8

Matrix representation of D₄₀⋊₁₄C₄ ►in GL₆(𝔽₂₄₁)

91	0	0	0	0	0
195	98	0	0	0	0
0	0	222	222	0	0
0	0	19	222	0	0
0	0	0	0	19	222
0	0	0	0	19	19

71	204	0	0	0	0
32	170	0	0	0	0
0	0	0	0	19	222
0	0	0	0	19	19
0	0	222	222	0	0
0	0	19	222	0	0

64	0	0	0	0	0
0	64	0	0	0	0
0	0	1	0	0	0
0	0	0	240	0	0
0	0	0	0	222	19
0	0	0	0	19	19

G:=sub<GL(6,GF(241))| [91,195,0,0,0,0,0,98,0,0,0,0,0,0,222,19,0,0,0,0,222,222,0,0,0,0,0,0,19,19,0,0,0,0,222,19],[71,32,0,0,0,0,204,170,0,0,0,0,0,0,0,0,222,19,0,0,0,0,222,222,0,0,19,19,0,0,0,0,222,19,0,0],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,222,19,0,0,0,0,19,19] >;

D₄₀⋊₁₄C₄ in GAP, Magma, Sage, TeX

D_{40}\rtimes_{14}C_4

% in TeX

G:=Group("D40:14C4");

// GroupNames label

G:=SmallGroup(320,46);

// by ID

G=gap.SmallGroup(320,46);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,675,794,192,1684,851,102,12550]);

// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;

// generators/relations

G = D40⋊14C4 order 320 = 26·5

8th semidirect product of D40 and C4 acting via C4/C2=C2

G = D₄₀⋊₁₄C₄ order 320 = 2⁶·5

8^th semidirect product of D₄₀ and C₄ acting via C₄/C₂=C₂