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## G = Q8×C13⋊C3order 312 = 23·3·13

### Direct product of Q8 and C13⋊C3

Aliases: Q8×C13⋊C3, C52.3C6, C132(C3×Q8), (Q8×C13)⋊3C3, C26.8(C2×C6), C4.(C2×C13⋊C3), (C4×C13⋊C3).3C2, C2.3(C22×C13⋊C3), (C2×C13⋊C3).8C22, SmallGroup(312,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — Q8×C13⋊C3
 Chief series C1 — C13 — C26 — C2×C13⋊C3 — C4×C13⋊C3 — Q8×C13⋊C3
 Lower central C13 — C26 — Q8×C13⋊C3
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C13⋊C3
G = < a,b,c,d | a4=c13=d3=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

Smallest permutation representation of Q8×C13⋊C3
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 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(67 69 75)(68 72 71)(70 78 76)(73 74 77)(80 82 88)(81 85 84)(83 91 89)(86 87 90)(93 95 101)(94 98 97)(96 104 102)(99 100 103)

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

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

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

35 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 6A 6B 12A ··· 12F 13A 13B 13C 13D 26A 26B 26C 26D 52A ··· 52L order 1 2 3 3 4 4 4 6 6 12 ··· 12 13 13 13 13 26 26 26 26 52 ··· 52 size 1 1 13 13 2 2 2 13 13 26 ··· 26 3 3 3 3 3 3 3 3 6 ··· 6

35 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + - image C1 C2 C3 C6 Q8 C3×Q8 C13⋊C3 C2×C13⋊C3 Q8×C13⋊C3 kernel Q8×C13⋊C3 C4×C13⋊C3 Q8×C13 C52 C13⋊C3 C13 Q8 C4 C1 # reps 1 3 2 6 1 2 4 12 4

Matrix representation of Q8×C13⋊C3 in GL5(𝔽157)

 156 57 0 0 0 22 1 0 0 0 0 0 156 0 0 0 0 0 156 0 0 0 0 0 156
,
 145 33 0 0 0 86 12 0 0 0 0 0 156 0 0 0 0 0 156 0 0 0 0 0 156
,
 1 0 0 0 0 0 1 0 0 0 0 0 119 52 1 0 0 1 0 0 0 0 0 1 0
,
 144 0 0 0 0 0 144 0 0 0 0 0 1 0 0 0 0 104 118 52 0 0 73 53 38

G:=sub<GL(5,GF(157))| [156,22,0,0,0,57,1,0,0,0,0,0,156,0,0,0,0,0,156,0,0,0,0,0,156],[145,86,0,0,0,33,12,0,0,0,0,0,156,0,0,0,0,0,156,0,0,0,0,0,156],[1,0,0,0,0,0,1,0,0,0,0,0,119,1,0,0,0,52,0,1,0,0,1,0,0],[144,0,0,0,0,0,144,0,0,0,0,0,1,104,73,0,0,0,118,53,0,0,0,52,38] >;

Q8×C13⋊C3 in GAP, Magma, Sage, TeX

Q_8\times C_{13}\rtimes C_3
% in TeX

G:=Group("Q8xC13:C3");
// GroupNames label

G:=SmallGroup(312,24);
// by ID

G=gap.SmallGroup(312,24);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,60,141,66,464]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^13=d^3=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^9>;
// generators/relations

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