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G = Q8xC12order 96 = 25·3

Direct product of C12 and Q8

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

Aliases: Q8xC12, C42.3C6, C4:C4.6C6, C12o3(C4:C4), C2.2(C6xQ8), C4.4(C2xC12), (C4xC12).9C2, (C2xQ8).7C6, C6.19(C2xQ8), C12.31(C2xC4), (C6xQ8).10C2, C6.40(C4oD4), C2.5(C22xC12), (C2xC6).74C23, C6.33(C22xC4), C22.8(C22xC6), (C2xC12).122C22, C4o3(C3xC4:C4), C12o3(C3xC4:C4), C2.3(C3xC4oD4), (C3xC4:C4).13C2, (C2xC4).16(C2xC6), SmallGroup(96,166)

Series: Derived Chief Lower central Upper central

C1C2 — Q8xC12
C1C2C22C2xC6C2xC12C3xC4:C4 — Q8xC12
C1C2 — Q8xC12
C1C2xC12 — Q8xC12

Generators and relations for Q8xC12
 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, C3, C4, C4, C22, C6, C2xC4, C2xC4, Q8, C12, C12, C2xC6, C42, C4:C4, C2xQ8, C2xC12, C2xC12, C3xQ8, C4xQ8, C4xC12, C3xC4:C4, C6xQ8, Q8xC12
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, Q8, C23, C12, C2xC6, C22xC4, C2xQ8, C4oD4, C2xC12, C3xQ8, C22xC6, C4xQ8, C22xC12, C6xQ8, C3xC4oD4, Q8xC12

Smallest permutation representation of Q8xC12
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 48 83 21)(2 37 84 22)(3 38 73 23)(4 39 74 24)(5 40 75 13)(6 41 76 14)(7 42 77 15)(8 43 78 16)(9 44 79 17)(10 45 80 18)(11 46 81 19)(12 47 82 20)(25 94 72 57)(26 95 61 58)(27 96 62 59)(28 85 63 60)(29 86 64 49)(30 87 65 50)(31 88 66 51)(32 89 67 52)(33 90 68 53)(34 91 69 54)(35 92 70 55)(36 93 71 56)
(1 95 83 58)(2 96 84 59)(3 85 73 60)(4 86 74 49)(5 87 75 50)(6 88 76 51)(7 89 77 52)(8 90 78 53)(9 91 79 54)(10 92 80 55)(11 93 81 56)(12 94 82 57)(13 65 40 30)(14 66 41 31)(15 67 42 32)(16 68 43 33)(17 69 44 34)(18 70 45 35)(19 71 46 36)(20 72 47 25)(21 61 48 26)(22 62 37 27)(23 63 38 28)(24 64 39 29)

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

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

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

Q8xC12 is a maximal subgroup of
C12.26Q16  Q8:4Dic6  Q8:5Dic6  Q8.5Dic6  C42.210D6  C42.56D6  Q8:2D12  Q8.6D12  C42.59D6  C12:7Q16  Dic6:10Q8  C42.122D6  Q8:6Dic6  Q8:7Dic6  C42.125D6  C42.126D6  Q8:6D12  Q8:7D12  C42.232D6  D12:10Q8  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++++-
imageC1C2C2C2C3C4C6C6C6C12Q8C4oD4C3xQ8C3xC4oD4
kernelQ8xC12C4xC12C3xC4:C4C6xQ8C4xQ8C3xQ8C42C4:C4C2xQ8Q8C12C6C4C2
# reps133128662162244

Matrix representation of Q8xC12 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] >;

Q8xC12 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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