Copied to
clipboard

G = C2xC6.D8order 192 = 26·3

Direct product of C2 and C6.D8

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

Aliases: C2xC6.D8, C4:C4:35D6, (C2xD12):9C4, (C2xC6).39D8, C6.48(C2xD8), D12:19(C2xC4), C4.7(D6:C4), C4.59(C2xD12), C6:1(D4:C4), C12.139(C2xD4), (C2xC12).133D4, (C2xC4).139D12, (C2xC6).40SD16, C6.66(C2xSD16), C12.59(C22xC4), (C22xC6).183D4, (C22xC4).345D6, C12.19(C22:C4), (C2xC12).318C23, C22.45(D6:C4), C22.20(D4:S3), (C22xD12).10C2, (C2xD12).232C22, C23.104(C3:D4), (C22xC12).133C22, C22.11(Q8:2S3), (C6xC4:C4):1C2, (C2xC4:C4):1S3, C4.48(S3xC2xC4), C3:2(C2xD4:C4), (C22xC3:C8):1C2, C2.2(C2xD4:S3), (C2xC3:C8):31C22, (C2xC4).75(C4xS3), C2.11(C2xD6:C4), (C3xC4:C4):40C22, (C2xC12).77(C2xC4), (C2xC6).438(C2xD4), C6.38(C2xC22:C4), C2.2(C2xQ8:2S3), C22.57(C2xC3:D4), (C2xC4).124(C3:D4), (C2xC6).57(C22:C4), (C2xC4).418(C22xS3), SmallGroup(192,524)

Series: Derived Chief Lower central Upper central

C1C12 — C2xC6.D8
C1C3C6C2xC6C2xC12C2xD12C22xD12 — C2xC6.D8
C3C6C12 — C2xC6.D8
C1C23C22xC4C2xC4:C4

Generators and relations for C2xC6.D8
 G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 696 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C12, C12, C12, D6, C2xC6, C2xC6, C4:C4, C4:C4, C2xC8, C22xC4, C22xC4, C2xD4, C24, C3:C8, D12, D12, C2xC12, C2xC12, C2xC12, C22xS3, C22xC6, D4:C4, C2xC4:C4, C22xC8, C22xD4, C2xC3:C8, C2xC3:C8, C3xC4:C4, C3xC4:C4, C2xD12, C2xD12, C22xC12, C22xC12, S3xC23, C2xD4:C4, C6.D8, C22xC3:C8, C6xC4:C4, C22xD12, C2xC6.D8
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22:C4, D8, SD16, C22xC4, C2xD4, C4xS3, D12, C3:D4, C22xS3, D4:C4, C2xC22:C4, C2xD8, C2xSD16, D6:C4, D4:S3, Q8:2S3, S3xC2xC4, C2xD12, C2xC3:D4, C2xD4:C4, C6.D8, C2xD6:C4, C2xD4:S3, C2xQ8:2S3, C2xC6.D8

Smallest permutation representation of C2xC6.D8
On 96 points
Generators in S96
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 65)(48 66)(73 85)(74 86)(75 87)(76 88)(77 81)(78 82)(79 83)(80 84)
(1 83 10 47 62 53)(2 54 63 48 11 84)(3 85 12 41 64 55)(4 56 57 42 13 86)(5 87 14 43 58 49)(6 50 59 44 15 88)(7 81 16 45 60 51)(8 52 61 46 9 82)(17 75 94 69 32 34)(18 35 25 70 95 76)(19 77 96 71 26 36)(20 37 27 72 89 78)(21 79 90 65 28 38)(22 39 29 66 91 80)(23 73 92 67 30 40)(24 33 31 68 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)
(1 17)(2 68)(3 23)(4 66)(5 21)(6 72)(7 19)(8 70)(9 76)(10 32)(11 74)(12 30)(13 80)(14 28)(15 78)(16 26)(18 46)(20 44)(22 42)(24 48)(25 52)(27 50)(29 56)(31 54)(33 63)(34 83)(35 61)(36 81)(37 59)(38 87)(39 57)(40 85)(41 67)(43 65)(45 71)(47 69)(49 79)(51 77)(53 75)(55 73)(58 90)(60 96)(62 94)(64 92)(82 95)(84 93)(86 91)(88 89)

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E4F4G4H6A···6G8A···8H12A···12L
order12···222223444444446···68···812···12
size11···1121212122222244442···26···64···4

48 irreducible representations

dim1111112222222222244
type++++++++++++++
imageC1C2C2C2C2C4S3D4D4D6D6D8SD16C4xS3D12C3:D4C3:D4D4:S3Q8:2S3
kernelC2xC6.D8C6.D8C22xC3:C8C6xC4:C4C22xD12C2xD12C2xC4:C4C2xC12C22xC6C4:C4C22xC4C2xC6C2xC6C2xC4C2xC4C2xC4C23C22C22
# reps1411181312144442222

Matrix representation of C2xC6.D8 in GL6(F73)

7200000
0720000
0072000
0007200
0000720
0000072
,
0720000
110000
0072100
0072000
000010
000001
,
0460000
4600000
0007200
0072000
00001616
00005716
,
0720000
7200000
0007200
0072000
000001
000010

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,16,57,0,0,0,0,16,16],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2xC6.D8 in GAP, Magma, Sage, TeX

C_2\times C_6.D_8
% in TeX

G:=Group("C2xC6.D8");
// GroupNames label

G:=SmallGroup(192,524);
// by ID

G=gap.SmallGroup(192,524);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1684,438,102,6278]);
// Polycyclic

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

׿
x
:
Z
F
o
wr
Q
<