C2xC52:C4 - GroupNames

G = C₂×C₅₂⋊C₄ order 416 = 2⁵·13

Direct product of C₂ and C₅₂⋊C₄

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

Aliases: C₂×C₅₂⋊C₄, D₂₆.₄Q₈, D₂₆.₁₁D₄, D₂₆.₁₀C₂³, C₂₆⋊(C₄⋊C₄), D₁₃⋊(C₄⋊C₄), C₅₂⋊₂(C₂×C₄), (C₂×C₅₂)⋊₃C₄, (C₄×D₁₃)⋊₄C₄, D₁₃.₁(C₂×D₄), D₁₃.₁(C₂×Q₈), (C₂×Dic₁₃)⋊₈C₄, Dic₁₃⋊₇(C₂×C₄), D₂₆.₁₆(C₂×C₄), C₂₆.₅(C₂²×C₄), (C₄×D₁₃).₃₀C₂², (C₂²×D₁₃).₃₇C₂², C₁₃⋊(C₂×C₄⋊C₄), C₄⋊₂(C₂×C₁₃⋊C₄), (C₂×C₄)⋊₃(C₁₃⋊C₄), (C₂×C₄×D₁₃).₁₄C₂, C₂.₆(C₂²×C₁₃⋊C₄), (C₂×C₂₆).₁₇(C₂×C₄), (C₂×C₁₃⋊C₄).₁C₂², (C₂²×C₁₃⋊C₄).₂C₂, C₂².₁₈(C₂×C₁₃⋊C₄), SmallGroup(416,203)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₆ — C₂×C₅₂⋊C₄

Chief series

C₁ — C₁₃ — D₁₃ — D₂₆ — C₂×C₁₃⋊C₄ — C₂²×C₁₃⋊C₄ — C₂×C₅₂⋊C₄

Lower central

C₁₃ — C₂₆ — C₂×C₅₂⋊C₄

Upper central

C₁ — C₂² — C₂×C₄

Generators and relations for C₂×C₅₂⋊C₄
G = < a,b,c | a²=b⁵²=c⁴=1, ab=ba, ac=ca, cbc^-1=b³¹ >

Subgroups: 660 in 92 conjugacy classes, 46 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₁₃, C₄⋊C₄, C₂²×C₄, D₁₃, D₁₃, C₂₆, C₂₆, C₂×C₄⋊C₄, Dic₁₃, C₅₂, C₁₃⋊C₄, D₂₆, D₂₆, C₂×C₂₆, C₄×D₁₃, C₂×Dic₁₃, C₂×C₅₂, C₂×C₁₃⋊C₄, C₂×C₁₃⋊C₄, C₂²×D₁₃, C₅₂⋊C₄, C₂×C₄×D₁₃, C₂²×C₁₃⋊C₄, C₂×C₅₂⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×C₄⋊C₄, C₁₃⋊C₄, C₂×C₁₃⋊C₄, C₅₂⋊C₄, C₂²×C₁₃⋊C₄, C₂×C₅₂⋊C₄

Smallest permutation representation of C₂×C₅₂⋊C₄
►On 104 points

Generators in S₁₀₄

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4	4	4	4
type	+	+	+	+				+	-	+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	C₁₃⋊C₄	C₂×C₁₃⋊C₄	C₂×C₁₃⋊C₄	C₅₂⋊C₄
kernel	C₂×C₅₂⋊C₄	C₅₂⋊C₄	C₂×C₄×D₁₃	C₂²×C₁₃⋊C₄	C₄×D₁₃	C₂×Dic₁₃	C₂×C₅₂	D₂₆	D₂₆	C₂×C₄	C₄	C₂²	C₂
# reps	1	4	1	2	4	2	2	2	2	3	6	3	12

Matrix representation of C₂×C₅₂⋊C₄ ►in GL₆(𝔽₅₃)

52	0	0	0	0	0
0	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

42	13	0	0	0	0
11	11	0	0	0	0
0	0	46	37	32	16
0	0	17	52	47	52
0	0	24	46	39	23
0	0	1	33	0	21

12	34	0	0	0	0
16	41	0	0	0	0
0	0	52	38	15	1
0	0	32	4	45	30
0	0	41	28	18	28
0	0	15	4	9	32

G:=sub<GL(6,GF(53))| [52,0,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,11,0,0,0,0,13,11,0,0,0,0,0,0,46,17,24,1,0,0,37,52,46,33,0,0,32,47,39,0,0,0,16,52,23,21],[12,16,0,0,0,0,34,41,0,0,0,0,0,0,52,32,41,15,0,0,38,4,28,4,0,0,15,45,18,9,0,0,1,30,28,32] >;

C₂×C₅₂⋊C₄ in GAP, Magma, Sage, TeX

C_2\times C_{52}\rtimes C_4

% in TeX

G:=Group("C2xC52:C4");

// GroupNames label

G:=SmallGroup(416,203);

// by ID

G=gap.SmallGroup(416,203);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,86,9221,1751]);

// Polycyclic

G:=Group<a,b,c|a^2=b^52=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^31>;

// generators/relations

G = C2×C52⋊C4 order 416 = 25·13

Direct product of C2 and C52⋊C4

G = C₂×C₅₂⋊C₄ order 416 = 2⁵·13

Direct product of C₂ and C₅₂⋊C₄