C5xD8:2C4 - GroupNames

G = C₅xD₈:₂C₄ order 320 = 2⁶·5

Direct product of C₅ and D₈:₂C₄

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C₅xD₈:₂C₄, D₈:₂C₂₀, Q₁₆:₂C₂₀, C₄₀.₁₀₂D₄, M₅(2):₅C₁₀, C₂₀.₄₃SD₁₆, (C₅xD₈):₁₄C₄, C₈.₁(C₂xC₂₀), C₄.Q₈:₁C₁₀, C₈.₂₂(C₅xD₄), C₄₀.₈₃(C₂xC₄), (C₅xQ₁₆):₁₄C₄, C₄oD₈.₂C₁₀, (C₂xC₁₀).₂₅D₈, C₄.₈(C₅xSD₁₆), C₂².₃(C₅xD₈), (C₂xC₂₀).₂₇₉D₄, (C₅xM₅(2)):₁₃C₂, (C₂xC₄₀).₂₆₅C₂², C₁₀.₅₅(D₄:C₄), C₂₀.₁₁₉(C₂²:C₄), (C₅xC₄oD₈).₇C₂, (C₅xC₄.Q₈):₁₀C₂, (C₂xC₄).₁₀(C₅xD₄), C₄.₄(C₅xC₂²:C₄), (C₂xC₈).₁₂(C₂xC₁₀), C₂.₉(C₅xD₄:C₄), SmallGroup(320,165)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₈ — C₅xD₈:₂C₄

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xC₈ — C₂xC₄₀ — C₅xC₄.Q₈ — C₅xD₈:₂C₄

Lower central

C₁ — C₂ — C₄ — C₈ — C₅xD₈:₂C₄

Upper central

C₁ — C₁₀ — C₂xC₂₀ — C₂xC₄₀ — C₅xD₈:₂C₄

Generators and relations for C₅xD₈:₂C₄
G = < a,b,c,d | a⁵=b⁸=c²=d⁴=1, ab=ba, ac=ca, ad=da, cbc=b^-1, dbd^-1=b³, dcd^-1=b⁵c >

Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₁₀, C₁₀, C₁₆, C₄:C₄, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₂₀, C₂₀, C₂xC₁₀, C₂xC₁₀, C₄.Q₈, M₅(2), C₄oD₈, C₄₀, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xQ₈, D₈:₂C₄, C₈₀, C₅xC₄:C₄, C₂xC₄₀, C₅xD₈, C₅xSD₁₆, C₅xQ₁₆, C₅xC₄oD₄, C₅xC₄.Q₈, C₅xM₅(2), C₅xC₄oD₈, C₅xD₈:₂C₄
Quotients: C₁, C₂, C₄, C₂², C₅, C₂xC₄, D₄, C₁₀, C₂²:C₄, D₈, SD₁₆, C₂₀, C₂xC₁₀, D₄:C₄, C₂xC₂₀, C₅xD₄, D₈:₂C₄, C₅xC₂²:C₄, C₅xD₈, C₅xSD₁₆, C₅xD₄:C₄, C₅xD₈:₂C₄

Smallest permutation representation of C₅xD₈:₂C₄
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

80 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	5A	5B	5C	5D	8A	8B	8C	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	10K	10L	16A	16B	16C	16D	20A	···	20H	20I	···	20T	40A	···	40H	40I	40J	40K	40L	80A	···	80P
order	1	2	2	2	4	4	4	4	4	5	5	5	5	8	8	8	10	10	10	10	10	10	10	10	10	10	10	10	16	16	16	16	20	···	20	20	···	20	40	···	40	40	40	40	40	80	···	80
size	1	1	2	8	2	2	8	8	8	1	1	1	1	2	2	4	1	1	1	1	2	2	2	2	8	8	8	8	4	4	4	4	2	···	2	8	···	8	2	···	2	4	4	4	4	4	···	4

80 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+									+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₅	C₁₀	C₁₀	C₁₀	C₂₀	C₂₀	D₄	D₄	SD₁₆	D₈	C₅xD₄	C₅xD₄	C₅xSD₁₆	C₅xD₈	D₈:₂C₄	C₅xD₈:₂C₄
kernel	C₅xD₈:₂C₄	C₅xC₄.Q₈	C₅xM₅(2)	C₅xC₄oD₈	C₅xD₈	C₅xQ₁₆	D₈:₂C₄	C₄.Q₈	M₅(2)	C₄oD₈	D₈	Q₁₆	C₄₀	C₂xC₂₀	C₂₀	C₂xC₁₀	C₈	C₂xC₄	C₄	C₂²	C₅	C₁
# reps	1	1	1	1	2	2	4	4	4	4	8	8	1	1	2	2	4	4	8	8	2	8

Matrix representation of C₅xD₈:₂C₄ ►in GL₄(F₂₄₁) generated by

87	0	0	0
0	87	0	0
0	0	87	0
0	0	0	87

0	19	43	32
203	38	166	11
0	0	222	222
0	0	19	222

64	11	158	121
0	0	19	222
203	0	166	11
203	38	166	11

240	0	11	75
239	1	199	64
0	0	222	222
0	0	222	19

G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[0,203,0,0,19,38,0,0,43,166,222,19,32,11,222,222],[64,0,203,203,11,0,0,38,158,19,166,166,121,222,11,11],[240,239,0,0,0,1,0,0,11,199,222,222,75,64,222,19] >;

C₅xD₈:₂C₄ in GAP, Magma, Sage, TeX

C_5\times D_8\rtimes_2C_4

% in TeX

G:=Group("C5xD8:2C4");

// GroupNames label

G:=SmallGroup(320,165);

// by ID

G=gap.SmallGroup(320,165);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,3650,136,3511,10085,5052,124]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;

// generators/relations

G = C5xD8:2C4 order 320 = 26·5

Direct product of C5 and D8:2C4

G = C₅xD₈:₂C₄ order 320 = 2⁶·5

Direct product of C₅ and D₈:₂C₄