Copied to
clipboard

G = C3xC4oD16order 192 = 26·3

Direct product of C3 and C4oD16

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

Aliases: C3xC4oD16, D16:3C6, Q32:3C6, SD32:3C6, C12.71D8, C24.76D4, C48.23C22, C24.66C23, (C2xC16):6C6, C4oD8:1C6, (C2xC48):13C2, (C3xD16):7C2, C16.6(C2xC6), (C3xQ32):7C2, D8.2(C2xC6), C4.20(C3xD8), (C2xC6).12D8, C4.10(C6xD4), C6.87(C2xD8), C2.15(C6xD8), C8.13(C3xD4), (C3xSD32):7C2, C8.6(C22xC6), Q16.2(C2xC6), C22.1(C3xD8), C12.317(C2xD4), (C2xC12).429D4, (C3xD8).12C22, (C2xC24).412C22, (C3xQ16).14C22, (C3xC4oD8):8C2, (C2xC8).91(C2xC6), (C2xC4).85(C3xD4), SmallGroup(192,941)

Series: Derived Chief Lower central Upper central

C1C8 — C3xC4oD16
C1C2C4C8C24C3xD8C3xD16 — C3xC4oD16
C1C2C4C8 — C3xC4oD16
C1C12C2xC12C2xC24 — C3xC4oD16

Generators and relations for C3xC4oD16
 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, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C12, C12, C2xC6, C2xC6, C16, C2xC8, D8, SD16, Q16, C4oD4, C24, C2xC12, C2xC12, C3xD4, C3xQ8, C2xC16, D16, SD32, Q32, C4oD8, C48, C2xC24, C3xD8, C3xSD16, C3xQ16, C3xC4oD4, C4oD16, C2xC48, C3xD16, C3xSD32, C3xQ32, C3xC4oD8, C3xC4oD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, D8, C2xD4, C3xD4, C22xC6, C2xD8, C3xD8, C6xD4, C4oD16, C6xD8, C3xC4oD16

Smallest permutation representation of C3xC4oD16
On 96 points
Generators in S96
(1 24 72)(2 25 73)(3 26 74)(4 27 75)(5 28 76)(6 29 77)(7 30 78)(8 31 79)(9 32 80)(10 17 65)(11 18 66)(12 19 67)(13 20 68)(14 21 69)(15 22 70)(16 23 71)(33 50 85)(34 51 86)(35 52 87)(36 53 88)(37 54 89)(38 55 90)(39 56 91)(40 57 92)(41 58 93)(42 59 94)(43 60 95)(44 61 96)(45 62 81)(46 63 82)(47 64 83)(48 49 84)
(1 33 9 41)(2 34 10 42)(3 35 11 43)(4 36 12 44)(5 37 13 45)(6 38 14 46)(7 39 15 47)(8 40 16 48)(17 59 25 51)(18 60 26 52)(19 61 27 53)(20 62 28 54)(21 63 29 55)(22 64 30 56)(23 49 31 57)(24 50 32 58)(65 94 73 86)(66 95 74 87)(67 96 75 88)(68 81 76 89)(69 82 77 90)(70 83 78 91)(71 84 79 92)(72 85 80 93)
(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 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 32)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 50)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 80)(81 88)(82 87)(83 86)(84 85)(89 96)(90 95)(91 94)(92 93)

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim1111111111112222222222
type++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D8D8C3xD4C3xD4C3xD8C3xD8C4oD16C3xC4oD16
kernelC3xC4oD16C2xC48C3xD16C3xSD32C3xQ32C3xC4oD8C4oD16C2xC16D16SD32Q32C4oD8C24C2xC12C12C2xC6C8C2xC4C4C22C3C1
# reps11121222242411222244816

Matrix representation of C3xC4oD16 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] >;

C3xC4oD16 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

׿
x
:
Z
F
o
wr
Q
<