Copied to
clipboard

G = D12.23D4order 192 = 26·3

6th non-split extension by D12 of D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.23D4, C42.63D6, C4.50(S3xD4), (C4xD12):21C2, C12:C8:29C2, (C2xD4).47D6, C12.24(C2xD4), C4.4D4:1S3, (C2xQ8).61D6, (C2xC12).271D4, C3:5(D4.2D4), C12.68(C4oD4), C6.105(C4oD8), D4:Dic3:19C2, Q8:2Dic3:22C2, C4.2(D4:2S3), (C6xD4).63C22, (C6xQ8).55C22, C2.18(D4:D6), C2.11(D6:3D4), C6.102(C4:D4), C6.119(C8:C22), (C2xC12).375C23, (C4xC12).106C22, C2.24(Q8.13D6), (C2xD12).244C22, C4:Dic3.341C22, (C2xD4:S3).6C2, (C2xC6).506(C2xD4), (C3xC4.4D4):1C2, (C2xQ8:2S3):12C2, (C2xC4).61(C3:D4), (C2xC3:C8).121C22, (C2xC4).475(C22xS3), C22.181(C2xC3:D4), SmallGroup(192,616)

Series: Derived Chief Lower central Upper central

C1C2xC12 — D12.23D4
C1C3C6C12C2xC12C2xD12C4xD12 — D12.23D4
C3C6C2xC12 — D12.23D4
C1C22C42C4.4D4

Generators and relations for D12.23D4
 G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 384 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, D8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C3:C8, C4xS3, D12, D12, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xC6, D4:C4, Q8:C4, C4:C8, C4xD4, C4.4D4, C2xD8, C2xSD16, C2xC3:C8, C4:Dic3, D6:C4, D4:S3, Q8:2S3, C4xC12, C3xC22:C4, S3xC2xC4, C2xD12, C6xD4, C6xQ8, D4.2D4, C12:C8, D4:Dic3, Q8:2Dic3, C4xD12, C2xD4:S3, C2xQ8:2S3, C3xC4.4D4, D12.23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, C4oD8, C8:C22, S3xD4, D4:2S3, C2xC3:D4, D4.2D4, D6:3D4, D4:D6, Q8.13D6, D12.23D4

Smallest permutation representation of D12.23D4
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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 84)(10 83)(11 82)(12 81)(13 26)(14 25)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 65)(50 64)(51 63)(52 62)(53 61)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)
(1 64 81 51)(2 65 82 52)(3 66 83 53)(4 67 84 54)(5 68 73 55)(6 69 74 56)(7 70 75 57)(8 71 76 58)(9 72 77 59)(10 61 78 60)(11 62 79 49)(12 63 80 50)(13 42 30 95)(14 43 31 96)(15 44 32 85)(16 45 33 86)(17 46 34 87)(18 47 35 88)(19 48 36 89)(20 37 25 90)(21 38 26 91)(22 39 27 92)(23 40 28 93)(24 41 29 94)
(1 95 7 89)(2 90 8 96)(3 85 9 91)(4 92 10 86)(5 87 11 93)(6 94 12 88)(13 51 19 57)(14 58 20 52)(15 53 21 59)(16 60 22 54)(17 55 23 49)(18 50 24 56)(25 65 31 71)(26 72 32 66)(27 67 33 61)(28 62 34 68)(29 69 35 63)(30 64 36 70)(37 76 43 82)(38 83 44 77)(39 78 45 84)(40 73 46 79)(41 80 47 74)(42 75 48 81)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D12A···12F12G12H
order122222234444444466666888812···121212
size1111812122222248121222288121212124···488

33 irreducible representations

dim1111111122222222244444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C4oD4C3:D4C4oD8C8:C22S3xD4D4:2S3D4:D6Q8.13D6
kernelD12.23D4C12:C8D4:Dic3Q8:2Dic3C4xD12C2xD4:S3C2xQ8:2S3C3xC4.4D4C4.4D4D12C2xC12C42C2xD4C2xQ8C12C2xC4C6C6C4C4C2C2
# reps1111111112211124411122

Matrix representation of D12.23D4 in GL6(F73)

7200000
0720000
0007200
0017200
0000171
0000172
,
100000
0720000
0017200
0007200
0000722
000001
,
4600000
0270000
0072000
0007200
0000270
0000027
,
0270000
4600000
0072000
0007200
00001261
0000661

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,2,1],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,6,0,0,0,0,61,61] >;

D12.23D4 in GAP, Magma, Sage, TeX

D_{12}._{23}D_4
% in TeX

G:=Group("D12.23D4");
// GroupNames label

G:=SmallGroup(192,616);
// by ID

G=gap.SmallGroup(192,616);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<