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## G = Q8×C13⋊C4order 416 = 25·13

### Direct product of Q8 and C13⋊C4

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 [×2], C4 [×3], C4 [×8], C22, C2×C4 [×7], Q8, Q8 [×3], C13, C42 [×3], C4⋊C4 [×3], C2×Q8, D13 [×2], C26, C4×Q8, Dic13 [×3], C52 [×3], C13⋊C4 [×2], C13⋊C4 [×3], D26, Dic26 [×3], C4×D13 [×3], Q8×C13, C2×C13⋊C4, C2×C13⋊C4 [×3], C4×C13⋊C4 [×3], C52⋊C4 [×3], Q8×D13, Q8×C13⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, C22×C4, C2×Q8, C4○D4, C4×Q8, C13⋊C4, C2×C13⋊C4 [×3], 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

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