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## G = Q8×C12order 96 = 25·3

### Direct product of C12 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C12
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C4⋊C4 — Q8×C12
 Lower central C1 — C2 — Q8×C12
 Upper central C1 — C2×C12 — Q8×C12

Generators and relations for Q8×C12
G = < a,b,c | a12=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 70 conjugacy classes, 64 normal (16 characteristic)
C1, C2 [×3], C3, C4 [×8], C4 [×3], C22, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C12 [×8], C12 [×3], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C2×C12, C2×C12 [×6], C3×Q8 [×4], C4×Q8, C4×C12 [×3], C3×C4⋊C4 [×3], C6×Q8, Q8×C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], Q8 [×2], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×Q8, C4○D4, C2×C12 [×6], C3×Q8 [×2], C22×C6, C4×Q8, C22×C12, C6×Q8, C3×C4○D4, Q8×C12

Smallest permutation representation of Q8×C12
Regular action on 96 points
Generators in S96
(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)
(1 55 95 80)(2 56 96 81)(3 57 85 82)(4 58 86 83)(5 59 87 84)(6 60 88 73)(7 49 89 74)(8 50 90 75)(9 51 91 76)(10 52 92 77)(11 53 93 78)(12 54 94 79)(13 26 47 72)(14 27 48 61)(15 28 37 62)(16 29 38 63)(17 30 39 64)(18 31 40 65)(19 32 41 66)(20 33 42 67)(21 34 43 68)(22 35 44 69)(23 36 45 70)(24 25 46 71)
(1 48 95 14)(2 37 96 15)(3 38 85 16)(4 39 86 17)(5 40 87 18)(6 41 88 19)(7 42 89 20)(8 43 90 21)(9 44 91 22)(10 45 92 23)(11 46 93 24)(12 47 94 13)(25 78 71 53)(26 79 72 54)(27 80 61 55)(28 81 62 56)(29 82 63 57)(30 83 64 58)(31 84 65 59)(32 73 66 60)(33 74 67 49)(34 75 68 50)(35 76 69 51)(36 77 70 52)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E ··· 4P 6A ··· 6F 12A ··· 12H 12I ··· 12AF order 1 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 Q8 C4○D4 C3×Q8 C3×C4○D4 kernel Q8×C12 C4×C12 C3×C4⋊C4 C6×Q8 C4×Q8 C3×Q8 C42 C4⋊C4 C2×Q8 Q8 C12 C6 C4 C2 # reps 1 3 3 1 2 8 6 6 2 16 2 2 4 4

Matrix representation of Q8×C12 in GL4(𝔽13) generated by

 8 0 0 0 0 10 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 12 0 0 0 0 0 1 0 0 12 0
,
 12 0 0 0 0 12 0 0 0 0 0 8 0 0 8 0
G:=sub<GL(4,GF(13))| [8,0,0,0,0,10,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,0,8,0,0,8,0] >;

Q8×C12 in GAP, Magma, Sage, TeX

Q_8\times C_{12}
% in TeX

G:=Group("Q8xC12");
// GroupNames label

G:=SmallGroup(96,166);
// by ID

G=gap.SmallGroup(96,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,151,338]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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