Q8xC13:C4 - GroupNames

G = Q₈×C₁₃⋊C₄ order 416 = 2⁵·13

Direct product of Q₈ and C₁₃⋊C₄

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈×C₁₃⋊C₄, Dic₂₆⋊₃C₄, D₂₆.₁₃C₂³, C₁₃⋊(C₄×Q₈), C₅₂.₇(C₂×C₄), (Q₈×C₁₃)⋊₃C₄, D₁₃.₂(C₂×Q₈), (Q₈×D₁₃).₃C₂, C₅₂⋊C₄.₂C₂, D₁₃.₃(C₄○D₄), C₂₆.₁₀(C₂²×C₄), Dic₁₃.₂(C₂×C₄), (C₄×D₁₃).₁₄C₂², C₄.₇(C₂×C₁₃⋊C₄), (C₄×C₁₃⋊C₄).₁C₂, (C₂×C₁₃⋊C₄).₄C₂², C₂.₁₁(C₂²×C₁₃⋊C₄), SmallGroup(416,208)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₆ — Q₈×C₁₃⋊C₄

Chief series

C₁ — C₁₃ — D₁₃ — D₂₆ — C₂×C₁₃⋊C₄ — C₄×C₁₃⋊C₄ — Q₈×C₁₃⋊C₄

Lower central

C₁₃ — C₂₆ — Q₈×C₁₃⋊C₄

Upper central

C₁ — C₂ — Q₈

Generators and relations for Q₈×C₁₃⋊C₄
G = < a,b,c,d | a⁴=c¹³=d⁴=1, b²=a², bab^-1=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁵ >

Subgroups: 460 in 70 conjugacy classes, 38 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂×C₄, Q₈, Q₈, C₁₃, C₄², C₄⋊C₄, C₂×Q₈, D₁₃, C₂₆, C₄×Q₈, Dic₁₃, C₅₂, C₁₃⋊C₄, C₁₃⋊C₄, D₂₆, Dic₂₆, C₄×D₁₃, Q₈×C₁₃, C₂×C₁₃⋊C₄, C₂×C₁₃⋊C₄, C₄×C₁₃⋊C₄, C₅₂⋊C₄, Q₈×D₁₃, Q₈×C₁₃⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, Q₈, C₂³, C₂²×C₄, C₂×Q₈, C₄○D₄, C₄×Q₈, C₁₃⋊C₄, C₂×C₁₃⋊C₄, C₂²×C₁₃⋊C₄, Q₈×C₁₃⋊C₄

Smallest permutation representation of Q₈×C₁₃⋊C₄
►On 104 points

Generators in S₁₀₄

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

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

(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)

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

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

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

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

35 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	···	4P	13A	13B	13C	26A	26B	26C	52A	···	52I
order	1	2	2	2	4	4	4	4	4	4	4	4	···	4	13	13	13	26	26	26	52	···	52
size	1	1	13	13	2	2	2	13	13	13	13	26	···	26	4	4	4	4	4	4	8	···	8

35 irreducible representations

dim	1	1	1	1	1	1	2	2	4	4	8
type	+	+	+	+			-		+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	Q₈	C₄○D₄	C₁₃⋊C₄	C₂×C₁₃⋊C₄	Q₈×C₁₃⋊C₄
kernel	Q₈×C₁₃⋊C₄	C₄×C₁₃⋊C₄	C₅₂⋊C₄	Q₈×D₁₃	Dic₂₆	Q₈×C₁₃	C₁₃⋊C₄	D₁₃	Q₈	C₄	C₁
# reps	1	3	3	1	6	2	2	2	3	9	3

Matrix representation of Q₈×C₁₃⋊C₄ ►in GL₆(𝔽₅₃)

1	2	0	0	0	0
52	52	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

27	35	0	0	0	0
17	26	0	0	0	0
0	0	52	0	0	0
0	0	0	52	0	0
0	0	0	0	52	0
0	0	0	0	0	52

1	0	0	0	0	0
0	1	0	0	0	0
0	0	46	12	46	52
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0

52	0	0	0	0	0
0	52	0	0	0	0
0	0	1	0	0	0
0	0	7	48	15	8
0	0	12	38	4	45
0	0	0	0	1	0

G:=sub<GL(6,GF(53))| [1,52,0,0,0,0,2,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,17,0,0,0,0,35,26,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,1,0,0,0,0,12,0,1,0,0,0,46,0,0,1,0,0,52,0,0,0],[52,0,0,0,0,0,0,52,0,0,0,0,0,0,1,7,12,0,0,0,0,48,38,0,0,0,0,15,4,1,0,0,0,8,45,0] >;

Q₈×C₁₃⋊C₄ in GAP, Magma, Sage, TeX

Q_8\times C_{13}\rtimes C_4

% in TeX

G:=Group("Q8xC13:C4");

// GroupNames label

G:=SmallGroup(416,208);

// by ID

G=gap.SmallGroup(416,208);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,188,86,9221,1751]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^13=d^4=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;

// generators/relations

G = Q8×C13⋊C4 order 416 = 25·13

Direct product of Q8 and C13⋊C4

G = Q₈×C₁₃⋊C₄ order 416 = 2⁵·13

Direct product of Q₈ and C₁₃⋊C₄