Copied to
clipboard

G = C122D8order 192 = 26·3

2nd semidirect product of C12 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C122D8, D129D4, C42.73D6, C42(D4⋊S3), C41D42S3, C34(C4⋊D8), C4.53(S3×D4), C6.57(C2×D8), (C4×D12)⋊22C2, (C2×D4).55D6, C12⋊C831C2, C12.30(C2×D4), (C2×C12).147D4, C12.76(C4○D4), C4.3(D42S3), D4⋊Dic322C2, C6.94(C8⋊C22), (C6×D4).71C22, C2.12(D63D4), C6.103(C4⋊D4), (C4×C12).120C22, (C2×C12).390C23, C2.15(D126C22), (C2×D12).246C22, C4⋊Dic3.344C22, (C2×D4⋊S3)⋊13C2, (C3×C41D4)⋊1C2, C2.12(C2×D4⋊S3), (C2×C6).521(C2×D4), (C2×C3⋊C8).130C22, (C2×C4).185(C3⋊D4), (C2×C4).488(C22×S3), C22.194(C2×C3⋊D4), SmallGroup(192,631)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C122D8
C1C3C6C12C2×C12C2×D12C4×D12 — C122D8
C3C6C2×C12 — C122D8
C1C22C42C41D4

Generators and relations for C122D8
 G = < a,b,c | a12=b8=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >

Subgroups: 448 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], S3 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×11], C23 [×3], Dic3, C12 [×2], C12 [×2], C12, D6 [×4], C2×C6, C2×C6 [×6], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×2], C2×D4 [×3], C3⋊C8 [×2], C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C3×D4 [×8], C22×S3, C22×C6 [×2], D4⋊C4 [×2], C4⋊C8, C4×D4, C41D4, C2×D8 [×2], C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, D4⋊S3 [×4], C4×C12, S3×C2×C4, C2×D12, C6×D4 [×2], C6×D4 [×2], C4⋊D8, C12⋊C8, D4⋊Dic3 [×2], C4×D12, C2×D4⋊S3 [×2], C3×C41D4, C122D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], D8 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×D8, C8⋊C22, D4⋊S3 [×2], S3×D4, D42S3, C2×C3⋊D4, C4⋊D8, C2×D4⋊S3, D126C22, D63D4, C122D8

Smallest permutation representation of C122D8
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 47 49 92 72 16 36 79)(2 46 50 91 61 15 25 78)(3 45 51 90 62 14 26 77)(4 44 52 89 63 13 27 76)(5 43 53 88 64 24 28 75)(6 42 54 87 65 23 29 74)(7 41 55 86 66 22 30 73)(8 40 56 85 67 21 31 84)(9 39 57 96 68 20 32 83)(10 38 58 95 69 19 33 82)(11 37 59 94 70 18 34 81)(12 48 60 93 71 17 35 80)
(2 6)(3 11)(5 9)(8 12)(13 89)(14 94)(15 87)(16 92)(17 85)(18 90)(19 95)(20 88)(21 93)(22 86)(23 91)(24 96)(25 54)(26 59)(27 52)(28 57)(29 50)(30 55)(31 60)(32 53)(33 58)(34 51)(35 56)(36 49)(37 77)(38 82)(39 75)(40 80)(41 73)(42 78)(43 83)(44 76)(45 81)(46 74)(47 79)(48 84)(61 65)(62 70)(64 68)(67 71)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G6A6B6C6D6E6F6G8A8B8C8D12A···12F
order12222222344444446666666888812···12
size111188121222222412122228888121212124···4

33 irreducible representations

dim1111112222222244444
type+++++++++++++++-
imageC1C2C2C2C2C2S3D4D4D6D6D8C4○D4C3⋊D4C8⋊C22D4⋊S3S3×D4D42S3D126C22
kernelC122D8C12⋊C8D4⋊Dic3C4×D12C2×D4⋊S3C3×C41D4C41D4D12C2×C12C42C2×D4C12C12C2×C4C6C4C4C4C2
# reps1121211221242412112

Matrix representation of C122D8 in GL6(𝔽73)

7200000
0720000
000100
00727200
0000270
0000046
,
41410000
1600000
0072000
001100
0000072
000010
,
100000
72720000
001000
00727200
000010
0000072

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[41,16,0,0,0,0,41,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C122D8 in GAP, Magma, Sage, TeX

C_{12}\rtimes_2D_8
% in TeX

G:=Group("C12:2D8");
// GroupNames label

G:=SmallGroup(192,631);
// by ID

G=gap.SmallGroup(192,631);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽