Copied to
clipboard

G = C12.4C42order 192 = 26·3

4th non-split extension by C12 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.4C42, C23.10Dic6, M4(2).2Dic3, C4.4(C4×Dic3), C4.50(D6⋊C4), (C2×C4).129D12, (C2×C12).111D4, (C22×C6).8Q8, C4.Dic3.4C4, C6.6(C8.C4), C32(C4.C42), (C22×C4).342D6, (C2×M4(2)).9S3, (C3×M4(2)).4C4, C12.95(C22⋊C4), (C6×M4(2)).13C2, C22.4(C4⋊Dic3), C4.29(C6.D4), C2.3(C12.53D4), C6.17(C2.C42), C2.17(C6.C42), C22.22(Dic3⋊C4), (C22×C12).127C22, (C2×C3⋊C8).7C4, (C2×C6).9(C4⋊C4), (C22×C3⋊C8).2C2, (C2×C12).64(C2×C4), (C2×C4).141(C4×S3), (C2×C4).40(C2×Dic3), (C2×C4).270(C3⋊D4), (C2×C4.Dic3).12C2, SmallGroup(192,117)

Series: Derived Chief Lower central Upper central

C1C12 — C12.4C42
C1C3C6C12C2×C12C22×C12C22×C3⋊C8 — C12.4C42
C3C6C12 — C12.4C42
C1C2×C4C22×C4C2×M4(2)

Generators and relations for C12.4C42
 G = < a,b,c | a12=1, b4=c4=a6, bab-1=a5, cac-1=a7, cbc-1=a3b >

Subgroups: 168 in 90 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C22 [×3], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×6], C2×C4 [×6], C23, C12 [×4], C2×C6 [×3], C2×C6 [×2], C2×C8 [×6], M4(2) [×2], M4(2) [×4], C22×C4, C3⋊C8 [×4], C24 [×2], C2×C12 [×6], C22×C6, C22×C8, C2×M4(2), C2×M4(2), C2×C3⋊C8 [×2], C2×C3⋊C8 [×3], C4.Dic3 [×2], C4.Dic3, C2×C24, C3×M4(2) [×2], C3×M4(2), C22×C12, C4.C42, C22×C3⋊C8, C2×C4.Dic3, C6×M4(2), C12.4C42
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2.C42, C8.C4 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C4.C42, C12.53D4 [×2], C6.C42, C12.4C42

Smallest permutation representation of C12.4C42
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 38 30 59 7 44 36 53)(2 43 31 52 8 37 25 58)(3 48 32 57 9 42 26 51)(4 41 33 50 10 47 27 56)(5 46 34 55 11 40 28 49)(6 39 35 60 12 45 29 54)(13 82 90 66 19 76 96 72)(14 75 91 71 20 81 85 65)(15 80 92 64 21 74 86 70)(16 73 93 69 22 79 87 63)(17 78 94 62 23 84 88 68)(18 83 95 67 24 77 89 61)
(1 74 36 70 7 80 30 64)(2 81 25 65 8 75 31 71)(3 76 26 72 9 82 32 66)(4 83 27 67 10 77 33 61)(5 78 28 62 11 84 34 68)(6 73 29 69 12 79 35 63)(13 45 90 60 19 39 96 54)(14 40 91 55 20 46 85 49)(15 47 92 50 21 41 86 56)(16 42 93 57 22 48 87 51)(17 37 94 52 23 43 88 58)(18 44 95 59 24 38 89 53)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B6C6D6E8A8B8C8D8E···8L8M8N8O8P12A12B12C12D12E12F24A···24H
order12222234444446666688888···8888812121212121224···24
size11112221111222224444446···6121212122222444···4

48 irreducible representations

dim111111122222222224
type++++++--++-
imageC1C2C2C2C4C4C4S3D4Q8Dic3D6C4×S3D12C3⋊D4Dic6C8.C4C12.53D4
kernelC12.4C42C22×C3⋊C8C2×C4.Dic3C6×M4(2)C2×C3⋊C8C4.Dic3C3×M4(2)C2×M4(2)C2×C12C22×C6M4(2)C22×C4C2×C4C2×C4C2×C4C23C6C2
# reps111144413121424284

Matrix representation of C12.4C42 in GL4(𝔽73) generated by

0100
727200
00460
00127
,
715300
55200
00100
006751
,
46000
04600
002771
001346
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,46,1,0,0,0,27],[71,55,0,0,53,2,0,0,0,0,10,67,0,0,0,51],[46,0,0,0,0,46,0,0,0,0,27,13,0,0,71,46] >;

C12.4C42 in GAP, Magma, Sage, TeX

C_{12}._4C_4^2
% in TeX

G:=Group("C12.4C4^2");
// GroupNames label

G:=SmallGroup(192,117);
// by ID

G=gap.SmallGroup(192,117);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,184,570,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^4=c^4=a^6,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=a^3*b>;
// generators/relations

׿
×
𝔽