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G = Dic3×SD16order 192 = 26·3

Direct product of Dic3 and SD16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3×SD16, C37(C4×SD16), C2415(C2×C4), C85(C2×Dic3), C6.95(C4×D4), (C3×SD16)⋊3C4, Q82(C2×Dic3), (C8×Dic3)⋊9C2, (Q8×Dic3)⋊3C2, C8⋊Dic326C2, (C2×C8).261D6, C2.7(S3×SD16), (C2×D4).142D6, C6.59(C4○D8), (C2×Q8).136D6, (C2×SD16).5S3, (D4×Dic3).7C2, D4.1(C2×Dic3), (C6×SD16).3C2, C2.12(D4×Dic3), C6.42(C2×SD16), C12.95(C4○D4), Q82Dic324C2, C12.73(C22×C4), (C6×D4).87C22, C22.116(S3×D4), (C6×Q8).68C22, C4.30(D42S3), C4.3(C22×Dic3), (C2×C24).162C22, (C2×C12).438C23, (C2×Dic3).213D4, C2.7(Q8.7D6), D4⋊Dic3.14C2, C4⋊Dic3.168C22, (C4×Dic3).240C22, (C3×Q8)⋊6(C2×C4), (C3×D4).8(C2×C4), (C2×C6).350(C2×D4), (C2×C3⋊C8).272C22, (C2×C4).527(C22×S3), SmallGroup(192,720)

Series: Derived Chief Lower central Upper central

C1C12 — Dic3×SD16
C1C3C6C2×C6C2×C12C4×Dic3D4×Dic3 — Dic3×SD16
C3C6C12 — Dic3×SD16
C1C22C2×C4C2×SD16

Generators and relations for Dic3×SD16
 G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 296 in 122 conjugacy classes, 59 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×SD16, C22×Dic3, C6×D4, C6×Q8, C4×SD16, C8×Dic3, C8⋊Dic3, D4⋊Dic3, Q82Dic3, D4×Dic3, Q8×Dic3, C6×SD16, Dic3×SD16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, SD16, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C2×SD16, C4○D8, S3×D4, D42S3, C22×Dic3, C4×SD16, S3×SD16, Q8.7D6, D4×Dic3, Dic3×SD16

Smallest permutation representation of Dic3×SD16
On 96 points
Generators in S96
(1 75 30 19 53 66)(2 76 31 20 54 67)(3 77 32 21 55 68)(4 78 25 22 56 69)(5 79 26 23 49 70)(6 80 27 24 50 71)(7 73 28 17 51 72)(8 74 29 18 52 65)(9 37 47 63 87 92)(10 38 48 64 88 93)(11 39 41 57 81 94)(12 40 42 58 82 95)(13 33 43 59 83 96)(14 34 44 60 84 89)(15 35 45 61 85 90)(16 36 46 62 86 91)
(1 57 19 11)(2 58 20 12)(3 59 21 13)(4 60 22 14)(5 61 23 15)(6 62 24 16)(7 63 17 9)(8 64 18 10)(25 34 69 84)(26 35 70 85)(27 36 71 86)(28 37 72 87)(29 38 65 88)(30 39 66 81)(31 40 67 82)(32 33 68 83)(41 53 94 75)(42 54 95 76)(43 55 96 77)(44 56 89 78)(45 49 90 79)(46 50 91 80)(47 51 92 73)(48 52 93 74)
(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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(65 71)(67 69)(68 72)(73 77)(74 80)(76 78)(82 84)(83 87)(86 88)(89 95)(91 93)(92 96)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D
order12222234444444444444466666888888881212121224242424
size1111442223333446612121212222882222666644884444

42 irreducible representations

dim1111111112222222224444
type+++++++++++-++-+
imageC1C2C2C2C2C2C2C2C4S3D4D6Dic3D6D6SD16C4○D4C4○D8D42S3S3×D4S3×SD16Q8.7D6
kernelDic3×SD16C8×Dic3C8⋊Dic3D4⋊Dic3Q82Dic3D4×Dic3Q8×Dic3C6×SD16C3×SD16C2×SD16C2×Dic3C2×C8SD16C2×D4C2×Q8Dic3C12C6C4C22C2C2
# reps1111111181214114241122

Matrix representation of Dic3×SD16 in GL4(𝔽73) generated by

1000
0100
00172
0010
,
1000
0100
004627
00027
,
0600
616100
0010
0001
,
1100
07200
0010
0001
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,46,0,0,0,27,27],[0,61,0,0,6,61,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,72,0,0,0,0,1,0,0,0,0,1] >;

Dic3×SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm SD}_{16}
% in TeX

G:=Group("Dic3xSD16");
// GroupNames label

G:=SmallGroup(192,720);
// by ID

G=gap.SmallGroup(192,720);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,219,184,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,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=c^3>;
// generators/relations

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