C13xC4wrC2 - GroupNames

G = C₁₃xC₄wrC₂ order 416 = 2⁵·13

Direct product of C₁₃ and C₄wrC₂

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₁₃xC₄wrC₂, D₄:₂C₅₂, Q₈:₂C₅₂, C₄²:₃C₂₆, C₅₂.₆₆D₄, M₄(2):₄C₂₆, (C₄xC₅₂):₁₀C₂, (D₄xC₁₃):₈C₄, C₄.₃(C₂xC₅₂), (Q₈xC₁₃):₈C₄, C₅₂.₅₁(C₂xC₄), C₄oD₄.₁C₂₆, C₄.₁₇(D₄xC₁₃), (C₂xC₂₆).₂₂D₄, C₂².₃(D₄xC₁₃), C₂₆.₃₇(C₂²:C₄), (C₁₃xM₄(2)):₁₀C₂, (C₂xC₅₂).₁₁₆C₂², (C₂xC₄).₁₉(C₂xC₂₆), (C₁₃xC₄oD₄).₄C₂, C₂.₈(C₁₃xC₂²:C₄), SmallGroup(416,54)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₁₃xC₄wrC₂

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xC₅₂ — C₁₃xM₄(2) — C₁₃xC₄wrC₂

Lower central

C₁ — C₂ — C₄ — C₁₃xC₄wrC₂

Upper central

C₁ — C₅₂ — C₂xC₅₂ — C₁₃xC₄wrC₂

Generators and relations for C₁₃xC₄wrC₂
G = < a,b,c,d | a¹³=b⁴=c²=d⁴=1, ab=ba, ac=ca, ad=da, cbc=b^-1, bd=db, dcd^-1=b^-1c >

Subgroups: 68 in 44 conjugacy classes, 24 normal (all characteristic)
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₁₃, C₂²:C₄, C₂₆, C₄wrC₂, C₅₂, C₂xC₂₆, C₂xC₅₂, D₄xC₁₃, C₁₃xC₂²:C₄, C₁₃xC₄wrC₂

C₁

C₂

₂C₂

₄C₂

C₁₃

C₄

C₂²

C₄

₂C₄

₂C₂²

₂C₄

C₂₆

₂C₂₆

₄C₂₆

Q₈

C₂xC₄

D₄

₂D₄

₂C₂xC₄

₂C₈

C₂xC₂₆

C₅₂

₂C₂xC₂₆

₂C₅₂

C₄oD₄

C₄²

M₄(2)

Q₈xC₁₃

C₂xC₅₂

D₄xC₁₃

₂C₂xC₅₂

₂C₁₀₄

₂D₄xC₁₃

₂C₂xC₅₂

C₄wrC₂

C₁₃xM₄(2)

C₄xC₅₂

C₁₃xC₄oD₄

C₁₃xC₄wrC₂

Smallest permutation representation of C₁₃xC₄wrC₂
►On 104 points

Generators in S₁₀₄

(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 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)

(1 39 22 86)(2 27 23 87)(3 28 24 88)(4 29 25 89)(5 30 26 90)(6 31 14 91)(7 32 15 79)(8 33 16 80)(9 34 17 81)(10 35 18 82)(11 36 19 83)(12 37 20 84)(13 38 21 85)(40 55 70 95)(41 56 71 96)(42 57 72 97)(43 58 73 98)(44 59 74 99)(45 60 75 100)(46 61 76 101)(47 62 77 102)(48 63 78 103)(49 64 66 104)(50 65 67 92)(51 53 68 93)(52 54 69 94)

(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 92)(11 93)(12 94)(13 95)(14 61)(15 62)(16 63)(17 64)(18 65)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)

(1 86 22 39)(2 87 23 27)(3 88 24 28)(4 89 25 29)(5 90 26 30)(6 91 14 31)(7 79 15 32)(8 80 16 33)(9 81 17 34)(10 82 18 35)(11 83 19 36)(12 84 20 37)(13 85 21 38)

G:=sub<Sym(104)| (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,39,22,86)(2,27,23,87)(3,28,24,88)(4,29,25,89)(5,30,26,90)(6,31,14,91)(7,32,15,79)(8,33,16,80)(9,34,17,81)(10,35,18,82)(11,36,19,83)(12,37,20,84)(13,38,21,85)(40,55,70,95)(41,56,71,96)(42,57,72,97)(43,58,73,98)(44,59,74,99)(45,60,75,100)(46,61,76,101)(47,62,77,102)(48,63,78,103)(49,64,66,104)(50,65,67,92)(51,53,68,93)(52,54,69,94), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,92)(11,93)(12,94)(13,95)(14,61)(15,62)(16,63)(17,64)(18,65)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84), (1,86,22,39)(2,87,23,27)(3,88,24,28)(4,89,25,29)(5,90,26,30)(6,91,14,31)(7,79,15,32)(8,80,16,33)(9,81,17,34)(10,82,18,35)(11,83,19,36)(12,84,20,37)(13,85,21,38)>;

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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,39,22,86)(2,27,23,87)(3,28,24,88)(4,29,25,89)(5,30,26,90)(6,31,14,91)(7,32,15,79)(8,33,16,80)(9,34,17,81)(10,35,18,82)(11,36,19,83)(12,37,20,84)(13,38,21,85)(40,55,70,95)(41,56,71,96)(42,57,72,97)(43,58,73,98)(44,59,74,99)(45,60,75,100)(46,61,76,101)(47,62,77,102)(48,63,78,103)(49,64,66,104)(50,65,67,92)(51,53,68,93)(52,54,69,94), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,92)(11,93)(12,94)(13,95)(14,61)(15,62)(16,63)(17,64)(18,65)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84), (1,86,22,39)(2,87,23,27)(3,88,24,28)(4,89,25,29)(5,90,26,30)(6,91,14,31)(7,79,15,32)(8,80,16,33)(9,81,17,34)(10,82,18,35)(11,83,19,36)(12,84,20,37)(13,85,21,38) );

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),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,39,22,86),(2,27,23,87),(3,28,24,88),(4,29,25,89),(5,30,26,90),(6,31,14,91),(7,32,15,79),(8,33,16,80),(9,34,17,81),(10,35,18,82),(11,36,19,83),(12,37,20,84),(13,38,21,85),(40,55,70,95),(41,56,71,96),(42,57,72,97),(43,58,73,98),(44,59,74,99),(45,60,75,100),(46,61,76,101),(47,62,77,102),(48,63,78,103),(49,64,66,104),(50,65,67,92),(51,53,68,93),(52,54,69,94)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,92),(11,93),(12,94),(13,95),(14,61),(15,62),(16,63),(17,64),(18,65),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84)], [(1,86,22,39),(2,87,23,27),(3,88,24,28),(4,89,25,29),(5,90,26,30),(6,91,14,31),(7,79,15,32),(8,80,16,33),(9,81,17,34),(10,82,18,35),(11,83,19,36),(12,84,20,37),(13,85,21,38)]])

182 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	···	4G	4H	8A	8B	13A	···	13L	26A	···	26L	26M	···	26X	26Y	···	26AJ	52A	···	52X	52Y	···	52CF	52CG	···	52CR	104A	···	104X
order	1	2	2	2	4	4	4	···	4	4	8	8	13	···	13	26	···	26	26	···	26	26	···	26	52	···	52	52	···	52	52	···	52	104	···	104
size	1	1	2	4	1	1	2	···	2	4	4	4	1	···	1	1	···	1	2	···	2	4	···	4	1	···	1	2	···	2	4	···	4	4	···	4

182 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+	+									+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₁₃	C₂₆	C₂₆	C₂₆	C₅₂	C₅₂	D₄	D₄	C₄wrC₂	D₄xC₁₃	D₄xC₁₃	C₁₃xC₄wrC₂
kernel	C₁₃xC₄wrC₂	C₄xC₅₂	C₁₃xM₄(2)	C₁₃xC₄oD₄	D₄xC₁₃	Q₈xC₁₃	C₄wrC₂	C₄²	M₄(2)	C₄oD₄	D₄	Q₈	C₅₂	C₂xC₂₆	C₁₃	C₄	C₂²	C₁
# reps	1	1	1	1	2	2	12	12	12	12	24	24	1	1	4	12	12	48

Matrix representation of C₁₃xC₄wrC₂ ►in GL₂(F₃₁₃) generated by

277	0
0	277

288	0
0	25

0	25
288	0

25	0
0	1

G:=sub<GL(2,GF(313))| [277,0,0,277],[288,0,0,25],[0,288,25,0],[25,0,0,1] >;

C₁₃xC₄wrC₂ in GAP, Magma, Sage, TeX

C_{13}\times C_4\wr C_2

% in TeX

G:=Group("C13xC4wrC2");

// GroupNames label

G:=SmallGroup(416,54);

// by ID

G=gap.SmallGroup(416,54);

# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,3129,117,88]);

// Polycyclic

G:=Group<a,b,c,d|a^13=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;

// generators/relations

Export

Subgroup lattice of C₁₃xC₄wrC₂ in TeX

G = C13xC4wrC2 order 416 = 25·13

Direct product of C13 and C4wrC2

G = C₁₃xC₄wrC₂ order 416 = 2⁵·13

Direct product of C₁₃ and C₄wrC₂