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G = C3×C4○D16order 192 = 26·3

Direct product of C3 and C4○D16

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C3×C4○D16, D163C6, Q323C6, SD323C6, C12.71D8, C24.76D4, C48.23C22, C24.66C23, (C2×C16)⋊6C6, C4○D81C6, (C2×C48)⋊13C2, (C3×D16)⋊7C2, C16.6(C2×C6), (C3×Q32)⋊7C2, D8.2(C2×C6), C4.20(C3×D8), (C2×C6).12D8, C4.10(C6×D4), C6.87(C2×D8), C2.15(C6×D8), C8.13(C3×D4), (C3×SD32)⋊7C2, C8.6(C22×C6), Q16.2(C2×C6), C22.1(C3×D8), C12.317(C2×D4), (C2×C12).429D4, (C3×D8).12C22, (C2×C24).412C22, (C3×Q16).14C22, (C3×C4○D8)⋊8C2, (C2×C8).91(C2×C6), (C2×C4).85(C3×D4), SmallGroup(192,941)

Series: Derived Chief Lower central Upper central

C1C8 — C3×C4○D16
C1C2C4C8C24C3×D8C3×D16 — C3×C4○D16
C1C2C4C8 — C3×C4○D16
C1C12C2×C12C2×C24 — C3×C4○D16

Generators and relations for C3×C4○D16
 G = < a,b,c,d | a3=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >

Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C16 [×2], C2×C8, D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×4], C3×Q8 [×2], C2×C16, D16, SD32 [×2], Q32, C4○D8 [×2], C48 [×2], C2×C24, C3×D8 [×2], C3×SD16 [×2], C3×Q16 [×2], C3×C4○D4 [×2], C4○D16, C2×C48, C3×D16, C3×SD32 [×2], C3×Q32, C3×C4○D8 [×2], C3×C4○D16
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], D8 [×2], C2×D4, C3×D4 [×2], C22×C6, C2×D8, C3×D8 [×2], C6×D4, C4○D16, C6×D8, C3×C4○D16

Smallest permutation representation of C3×C4○D16
On 96 points
Generators in S96
(1 91 73)(2 92 74)(3 93 75)(4 94 76)(5 95 77)(6 96 78)(7 81 79)(8 82 80)(9 83 65)(10 84 66)(11 85 67)(12 86 68)(13 87 69)(14 88 70)(15 89 71)(16 90 72)(17 61 37)(18 62 38)(19 63 39)(20 64 40)(21 49 41)(22 50 42)(23 51 43)(24 52 44)(25 53 45)(26 54 46)(27 55 47)(28 56 48)(29 57 33)(30 58 34)(31 59 35)(32 60 36)
(1 57 9 49)(2 58 10 50)(3 59 11 51)(4 60 12 52)(5 61 13 53)(6 62 14 54)(7 63 15 55)(8 64 16 56)(17 69 25 77)(18 70 26 78)(19 71 27 79)(20 72 28 80)(21 73 29 65)(22 74 30 66)(23 75 31 67)(24 76 32 68)(33 83 41 91)(34 84 42 92)(35 85 43 93)(36 86 44 94)(37 87 45 95)(38 88 46 96)(39 89 47 81)(40 90 48 82)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 24)(18 23)(19 22)(20 21)(25 32)(26 31)(27 30)(28 29)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 84)(82 83)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B8C8D12A12B12C12D12E12F12G12H12I12J16A···16H24A···24H48A···48P
order1222233444446666666688881212121212121212121216···1624···2448···48
size11288111128811228888222211112288882···22···22···2

66 irreducible representations

dim1111111111112222222222
type++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D8D8C3×D4C3×D4C3×D8C3×D8C4○D16C3×C4○D16
kernelC3×C4○D16C2×C48C3×D16C3×SD32C3×Q32C3×C4○D8C4○D16C2×C16D16SD32Q32C4○D8C24C2×C12C12C2×C6C8C2×C4C4C22C3C1
# reps11121222242411222244816

Matrix representation of C3×C4○D16 in GL2(𝔽97) generated by

350
035
,
750
075
,
226
712
,
712
226
G:=sub<GL(2,GF(97))| [35,0,0,35],[75,0,0,75],[2,71,26,2],[71,2,2,26] >;

C3×C4○D16 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{16}
% in TeX

G:=Group("C3xC4oD16");
// GroupNames label

G:=SmallGroup(192,941);
// by ID

G=gap.SmallGroup(192,941);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,520,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations

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