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G = C12xSD16order 192 = 26·3

Direct product of C12 and SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C12xSD16, (C4xC8):11C6, C8:5(C2xC12), (C4xQ8):5C6, C24:27(C2xC4), (C4xC24):27C2, Q8:2(C2xC12), C4.Q8:14C6, (C4xD4).4C6, (Q8xC12):21C2, D4.1(C2xC12), C2.13(D4xC12), C6.115(C4xD4), C2.4(C6xSD16), Q8:C4:21C6, (D4xC12).19C2, D4:C4.8C6, (C2xC12).361D4, C42.71(C2xC6), (C2xSD16).5C6, C6.84(C2xSD16), C22.52(C6xD4), C6.117(C4oD8), C4.10(C22xC12), (C6xSD16).10C2, C12.257(C4oD4), (C4xC12).356C22, (C2xC24).437C22, (C2xC12).905C23, C12.155(C22xC4), (C6xD4).291C22, (C6xQ8).254C22, C2.4(C3xC4oD8), C4.2(C3xC4oD4), C4:C4.46(C2xC6), (C2xC8).66(C2xC6), (C3xQ8):13(C2xC4), (C3xC4.Q8):29C2, (C2xC4).51(C3xD4), (C3xD4).18(C2xC4), (C2xD4).49(C2xC6), (C2xC6).628(C2xD4), (C2xQ8).51(C2xC6), (C3xQ8:C4):44C2, (C2xC4).80(C22xC6), (C3xD4:C4).17C2, (C3xC4:C4).367C22, SmallGroup(192,871)

Series: Derived Chief Lower central Upper central

C1C4 — C12xSD16
C1C2C22C2xC4C2xC12C3xC4:C4C3xQ8:C4 — C12xSD16
C1C2C4 — C12xSD16
C1C2xC12C4xC12 — C12xSD16

Generators and relations for C12xSD16
 G = < a,b,c | a12=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 202 in 122 conjugacy classes, 74 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C12, C12, C12, C2xC6, C2xC6, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C24, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xC6, C4xC8, D4:C4, Q8:C4, C4.Q8, C4xD4, C4xQ8, C2xSD16, C4xC12, C4xC12, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xC24, C3xSD16, C22xC12, C6xD4, C6xQ8, C4xSD16, C4xC24, C3xD4:C4, C3xQ8:C4, C3xC4.Q8, D4xC12, Q8xC12, C6xSD16, C12xSD16
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C23, C12, C2xC6, SD16, C22xC4, C2xD4, C4oD4, C2xC12, C3xD4, C22xC6, C4xD4, C2xSD16, C4oD8, C3xSD16, C22xC12, C6xD4, C3xC4oD4, C4xSD16, D4xC12, C6xSD16, C3xC4oD8, C12xSD16

Smallest permutation representation of C12xSD16
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 59 17 70 42 85 76 26)(2 60 18 71 43 86 77 27)(3 49 19 72 44 87 78 28)(4 50 20 61 45 88 79 29)(5 51 21 62 46 89 80 30)(6 52 22 63 47 90 81 31)(7 53 23 64 48 91 82 32)(8 54 24 65 37 92 83 33)(9 55 13 66 38 93 84 34)(10 56 14 67 39 94 73 35)(11 57 15 68 40 95 74 36)(12 58 16 69 41 96 75 25)
(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 96)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(49 72)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F4G4H4I···4N6A···6F6G6H6I6J8A···8H12A···12H12I···12P12Q···12AB24A···24P
order12222233444444444···46···666668···812···1212···1212···1224···24
size11114411111122224···41···144442···21···12···24···42···2

84 irreducible representations

dim11111111111111111122222222
type+++++++++
imageC1C2C2C2C2C2C2C2C3C4C6C6C6C6C6C6C6C12D4SD16C4oD4C3xD4C4oD8C3xSD16C3xC4oD4C3xC4oD8
kernelC12xSD16C4xC24C3xD4:C4C3xQ8:C4C3xC4.Q8D4xC12Q8xC12C6xSD16C4xSD16C3xSD16C4xC8D4:C4Q8:C4C4.Q8C4xD4C4xQ8C2xSD16SD16C2xC12C12C12C2xC4C6C4C4C2
# reps111111112822222221624244848

Matrix representation of C12xSD16 in GL3(F73) generated by

7000
0270
0027
,
7200
0067
01212
,
7200
011
0072
G:=sub<GL(3,GF(73))| [70,0,0,0,27,0,0,0,27],[72,0,0,0,0,12,0,67,12],[72,0,0,0,1,0,0,1,72] >;

C12xSD16 in GAP, Magma, Sage, TeX

C_{12}\times {\rm SD}_{16}
% in TeX

G:=Group("C12xSD16");
// GroupNames label

G:=SmallGroup(192,871);
// by ID

G=gap.SmallGroup(192,871);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,268,4204,2111,172]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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