Copied to
clipboard

G = C12.38SD16order 192 = 26·3

4th non-split extension by C12 of SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:4Dic6, C42.45D6, C12.38SD16, (C3xD4):4Q8, (C4xD4).3S3, C4:C4.239D6, C12:C8:18C2, C3:4(D4:2Q8), (D4xC12).3C2, (C2xC12).58D4, C12.26(C2xQ8), C12:2Q8:15C2, (C2xD4).186D6, C4.10(C2xDic6), C6.51(C2xSD16), C12.46(C4oD4), C4.60(C4oD12), C2.7(D4:D6), C4.13(D4.S3), (C4xC12).79C22, C12.Q8:30C2, D4:Dic3.8C2, C6.62(C22:Q8), C6.107(C8:C22), (C2xC12).333C23, (C6xD4).228C22, C4:Dic3.137C22, C2.13(C12.48D4), C2.6(C2xD4.S3), (C2xC6).464(C2xD4), (C2xC3:C8).90C22, (C2xC4).244(C3:D4), (C3xC4:C4).270C22, (C2xC4).433(C22xS3), C22.147(C2xC3:D4), SmallGroup(192,567)

Series: Derived Chief Lower central Upper central

C1C2xC12 — C12.38SD16
C1C3C6C12C2xC12C4:Dic3C12:2Q8 — C12.38SD16
C3C6C2xC12 — C12.38SD16
C1C22C42C4xD4

Generators and relations for C12.38SD16
 G = < a,b,c | a12=b8=c2=1, bab-1=a-1, ac=ca, cbc=a6b3 >

Subgroups: 280 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, C12, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C22xC4, C2xD4, C2xQ8, C3:C8, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C22xC6, D4:C4, C4:C8, C4.Q8, C4xD4, C4:Q8, C2xC3:C8, C4:Dic3, C4:Dic3, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, C22xC12, C6xD4, D4:2Q8, C12:C8, C12.Q8, D4:Dic3, C12:2Q8, D4xC12, C12.38SD16
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2xD4, C2xQ8, C4oD4, Dic6, C3:D4, C22xS3, C22:Q8, C2xSD16, C8:C22, D4.S3, C2xDic6, C4oD12, C2xC3:D4, D4:2Q8, C12.48D4, C2xD4.S3, D4:D6, C12.38SD16

Smallest permutation representation of C12.38SD16
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 83 86 36 46 68 14 58)(2 82 87 35 47 67 15 57)(3 81 88 34 48 66 16 56)(4 80 89 33 37 65 17 55)(5 79 90 32 38 64 18 54)(6 78 91 31 39 63 19 53)(7 77 92 30 40 62 20 52)(8 76 93 29 41 61 21 51)(9 75 94 28 42 72 22 50)(10 74 95 27 43 71 23 49)(11 73 96 26 44 70 24 60)(12 84 85 25 45 69 13 59)
(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 73)(33 74)(34 75)(35 76)(36 77)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 61)(58 62)(59 63)(60 64)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E···12L
order1222223444444444666666688881212121212···12
size11114422222444242422244441212121222224···4

39 irreducible representations

dim11111122222222222444
type++++++++-+++-+-+
imageC1C2C2C2C2C2S3D4Q8D6D6D6SD16C4oD4C3:D4Dic6C4oD12C8:C22D4.S3D4:D6
kernelC12.38SD16C12:C8C12.Q8D4:Dic3C12:2Q8D4xC12C4xD4C2xC12C3xD4C42C4:C4C2xD4C12C12C2xC4D4C4C6C4C2
# reps11221112211142444122

Matrix representation of C12.38SD16 in GL4(F73) generated by

3000
04900
00720
00072
,
0100
72000
00040
004212
,
1000
0100
0010
006672
G:=sub<GL(4,GF(73))| [3,0,0,0,0,49,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,1,0,0,0,0,0,0,42,0,0,40,12],[1,0,0,0,0,1,0,0,0,0,1,66,0,0,0,72] >;

C12.38SD16 in GAP, Magma, Sage, TeX

C_{12}._{38}{\rm SD}_{16}
% in TeX

G:=Group("C12.38SD16");
// GroupNames label

G:=SmallGroup(192,567);
// by ID

G=gap.SmallGroup(192,567);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,253,120,254,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=a^6*b^3>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<