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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

Aliases: Q8×C12, C42.3C6, C4⋊C4.6C6, C123(C4⋊C4), C2.2(C6×Q8), C4.4(C2×C12), (C4×C12).9C2, (C2×Q8).7C6, C6.19(C2×Q8), C12.31(C2×C4), (C6×Q8).10C2, C6.40(C4○D4), C2.5(C22×C12), (C2×C6).74C23, C6.33(C22×C4), C22.8(C22×C6), (C2×C12).122C22, C43(C3×C4⋊C4), C123(C3×C4⋊C4), C2.3(C3×C4○D4), (C3×C4⋊C4).13C2, (C2×C4).16(C2×C6), SmallGroup(96,166)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C12
C1C2C22C2×C6C2×C12C3×C4⋊C4 — Q8×C12
C1C2 — Q8×C12
C1C2×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)])

Q8×C12 is a maximal subgroup of
C12.26Q16  Q84Dic6  Q85Dic6  Q8.5Dic6  C42.210D6  C42.56D6  Q82D12  Q8.6D12  C42.59D6  C127Q16  Dic610Q8  C42.122D6  Q86Dic6  Q87Dic6  C42.125D6  C42.126D6  Q86D12  Q87D12  C42.232D6  D1210Q8  C42.131D6  C42.132D6  C42.133D6  C42.134D6  C42.135D6  C42.136D6

60 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E···4P6A···6F12A···12H12I···12AF
order12223344444···46···612···1212···12
size11111111112···21···11···12···2

60 irreducible representations

dim11111111112222
type++++-
imageC1C2C2C2C3C4C6C6C6C12Q8C4○D4C3×Q8C3×C4○D4
kernelQ8×C12C4×C12C3×C4⋊C4C6×Q8C4×Q8C3×Q8C42C4⋊C4C2×Q8Q8C12C6C4C2
# reps133128662162244

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

8000
01000
0080
0008
,
1000
01200
0001
00120
,
12000
01200
0008
0080
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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