Copied to
clipboard

G = C12.(C4⋊C4)  order 192 = 26·3

7th non-split extension by C12 of C4⋊C4 acting via C4⋊C4/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.7(C4⋊C4), (C2×C12).5Q8, (C2×C4).12D12, C4.Dic32C4, (C2×C4).1Dic6, C4.32(D6⋊C4), (C2×C12).103D4, (C22×C12).3C4, (C2×C6).18C42, (C22×C4).72D6, C12.8(C22⋊C4), C4.7(Dic3⋊C4), C6.7(C4.D4), C22.9(C4×Dic3), (C22×C4).5Dic3, C31(C22.C42), C2.2(C12.D4), C6.7(C4.10D4), C22.9(C4⋊Dic3), C23.28(C2×Dic3), C2.2(C12.10D4), C6.6(C2.C42), C2.7(C6.C42), (C22×C12).119C22, C22.27(C6.D4), (C6×C4⋊C4).2C2, (C2×C4⋊C4).3S3, (C2×C4).18(C4×S3), (C2×C6).36(C4⋊C4), (C2×C12).55(C2×C4), (C2×C4.Dic3).6C2, (C2×C4).175(C3⋊D4), (C2×C6).88(C22⋊C4), (C22×C6).125(C2×C4), SmallGroup(192,89)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12.(C4⋊C4)
C1C3C6C12C2×C12C22×C12C2×C4.Dic3 — C12.(C4⋊C4)
C3C6C2×C6 — C12.(C4⋊C4)
C1C22C22×C4C2×C4⋊C4

Generators and relations for C12.(C4⋊C4)
 G = < a,b,c | a12=c4=1, b4=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b3 >

Subgroups: 200 in 98 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×2], C22 [×3], C22 [×2], C6 [×3], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C3⋊C8 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×C6, C2×C4⋊C4, C2×M4(2) [×2], C2×C3⋊C8 [×2], C4.Dic3 [×4], C4.Dic3 [×2], C3×C4⋊C4 [×2], C22×C12, C22×C12 [×2], C22.C42, C2×C4.Dic3 [×2], C6×C4⋊C4, C12.(C4⋊C4)
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, C4.D4, C4.10D4, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C22.C42, C6.C42, C12.D4, C12.10D4, C12.(C4⋊C4)

Smallest permutation representation of C12.(C4⋊C4)
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 65 29 95 7 71 35 89)(2 64 30 94 8 70 36 88)(3 63 31 93 9 69 25 87)(4 62 32 92 10 68 26 86)(5 61 33 91 11 67 27 85)(6 72 34 90 12 66 28 96)(13 41 57 75 19 47 51 81)(14 40 58 74 20 46 52 80)(15 39 59 73 21 45 53 79)(16 38 60 84 22 44 54 78)(17 37 49 83 23 43 55 77)(18 48 50 82 24 42 56 76)
(1 50 32 15)(2 57 33 22)(3 52 34 17)(4 59 35 24)(5 54 36 19)(6 49 25 14)(7 56 26 21)(8 51 27 16)(9 58 28 23)(10 53 29 18)(11 60 30 13)(12 55 31 20)(37 96 80 69)(38 91 81 64)(39 86 82 71)(40 93 83 66)(41 88 84 61)(42 95 73 68)(43 90 74 63)(44 85 75 70)(45 92 76 65)(46 87 77 72)(47 94 78 67)(48 89 79 62)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A···6G8A···8H12A···12L
order1222223444444446···68···812···12
size1111222222244442···212···124···4

42 irreducible representations

dim111112222222224444
type+++++--+-++-
imageC1C2C2C4C4S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4C4.D4C4.10D4C12.D4C12.10D4
kernelC12.(C4⋊C4)C2×C4.Dic3C6×C4⋊C4C4.Dic3C22×C12C2×C4⋊C4C2×C12C2×C12C22×C4C22×C4C2×C4C2×C4C2×C4C2×C4C6C6C2C2
# reps121841312124241122

Matrix representation of C12.(C4⋊C4) in GL6(𝔽73)

800000
0640000
0007200
001000
00203401
003920720
,
0720000
100000
0051120
00516302
0072286862
0054712210
,
2700000
0460000
009800
0086400
0072206465
002027659

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,64,0,0,0,0,0,0,0,1,20,39,0,0,72,0,34,20,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,5,51,72,54,0,0,11,63,28,71,0,0,2,0,68,22,0,0,0,2,62,10],[27,0,0,0,0,0,0,46,0,0,0,0,0,0,9,8,72,20,0,0,8,64,20,27,0,0,0,0,64,65,0,0,0,0,65,9] >;

C12.(C4⋊C4) in GAP, Magma, Sage, TeX

C_{12}.(C_4\rtimes C_4)
% in TeX

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

G:=SmallGroup(192,89);
// by ID

G=gap.SmallGroup(192,89);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,184,1684,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=c^4=1,b^4=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b^3>;
// generators/relations

׿
×
𝔽