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G = C429D6order 192 = 26·3

7th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C429D6, C6.932+ 1+4, C4⋊C454D6, (C4×D12)⋊5C2, (C2×D12)⋊18C4, D1226(C2×C4), (C4×C12)⋊4C22, C42⋊C26S3, D6⋊C460C22, C2.1(D4○D12), C6.17(C23×C4), (C2×C6).65C24, Dic35D411C2, D6.4(C22×C4), C4⋊Dic382C22, C22⋊C4.125D6, (C22×C4).204D6, C12.120(C22×C4), (C2×C12).583C23, C32(C22.11C24), (C4×Dic3)⋊10C22, (C22×D12).17C2, C22.27(S3×C23), (C2×D12).255C22, C23.26D624C2, (S3×C23).35C22, (C22×C6).135C23, C23.163(C22×S3), (C22×S3).162C23, (C22×C12).225C22, (C2×Dic3).195C23, C6.D4.94C22, (C2×C4)⋊6(C4×S3), C4.58(S3×C2×C4), (C2×C12)⋊11(C2×C4), (S3×C2×C4)⋊43C22, C22.27(S3×C2×C4), C2.19(S3×C22×C4), (C3×C4⋊C4)⋊51C22, (S3×C22⋊C4)⋊25C2, (C22×S3)⋊7(C2×C4), (C3×C42⋊C2)⋊7C2, (C2×C6).21(C22×C4), (C2×C4).271(C22×S3), (C3×C22⋊C4).135C22, SmallGroup(192,1080)

Series: Derived Chief Lower central Upper central

C1C6 — C429D6
C1C3C6C2×C6C22×S3S3×C23C22×D12 — C429D6
C3C6 — C429D6
C1C22C42⋊C2

Generators and relations for C429D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 936 in 338 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×4], C4 [×8], C22, C22 [×2], C22 [×26], S3 [×8], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×16], C23, C23 [×16], Dic3 [×4], C12 [×4], C12 [×4], D6 [×8], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], C4×S3 [×8], D12 [×16], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×12], C22×S3 [×4], C22×C6, C2×C22⋊C4 [×4], C42⋊C2, C42⋊C2, C4×D4 [×8], C22×D4, C4×Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×8], C2×D12 [×12], C22×C12, S3×C23 [×2], C22.11C24, C4×D12 [×4], S3×C22⋊C4 [×4], Dic35D4 [×4], C23.26D6, C3×C42⋊C2, C22×D12, C429D6
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ 1+4 [×2], S3×C2×C4 [×6], S3×C23, C22.11C24, S3×C22×C4, D4○D12 [×2], C429D6

Smallest permutation representation of C429D6
On 48 points
Generators in S48
(1 38 14 36)(2 39 15 31)(3 40 16 32)(4 41 17 33)(5 42 18 34)(6 37 13 35)(7 27 44 20)(8 28 45 21)(9 29 46 22)(10 30 47 23)(11 25 48 24)(12 26 43 19)
(1 19 4 29)(2 27 5 23)(3 21 6 25)(7 34 47 31)(8 37 48 40)(9 36 43 33)(10 39 44 42)(11 32 45 35)(12 41 46 38)(13 24 16 28)(14 26 17 22)(15 20 18 30)
(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)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 36)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)

G:=sub<Sym(48)| (1,38,14,36)(2,39,15,31)(3,40,16,32)(4,41,17,33)(5,42,18,34)(6,37,13,35)(7,27,44,20)(8,28,45,21)(9,29,46,22)(10,30,47,23)(11,25,48,24)(12,26,43,19), (1,19,4,29)(2,27,5,23)(3,21,6,25)(7,34,47,31)(8,37,48,40)(9,36,43,33)(10,39,44,42)(11,32,45,35)(12,41,46,38)(13,24,16,28)(14,26,17,22)(15,20,18,30), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,36)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)>;

G:=Group( (1,38,14,36)(2,39,15,31)(3,40,16,32)(4,41,17,33)(5,42,18,34)(6,37,13,35)(7,27,44,20)(8,28,45,21)(9,29,46,22)(10,30,47,23)(11,25,48,24)(12,26,43,19), (1,19,4,29)(2,27,5,23)(3,21,6,25)(7,34,47,31)(8,37,48,40)(9,36,43,33)(10,39,44,42)(11,32,45,35)(12,41,46,38)(13,24,16,28)(14,26,17,22)(15,20,18,30), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,36)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43) );

G=PermutationGroup([(1,38,14,36),(2,39,15,31),(3,40,16,32),(4,41,17,33),(5,42,18,34),(6,37,13,35),(7,27,44,20),(8,28,45,21),(9,29,46,22),(10,30,47,23),(11,25,48,24),(12,26,43,19)], [(1,19,4,29),(2,27,5,23),(3,21,6,25),(7,34,47,31),(8,37,48,40),(9,36,43,33),(10,39,44,42),(11,32,45,35),(12,41,46,38),(13,24,16,28),(14,26,17,22),(15,20,18,30)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,36),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43)])

54 conjugacy classes

class 1 2A2B2C2D2E2F···2M 3 4A···4L4M···4T6A6B6C6D6E12A12B12C12D12E···12N
order1222222···234···44···4666661212121212···12
size1111226···622···26···62224422224···4

54 irreducible representations

dim1111111122222244
type++++++++++++++
imageC1C2C2C2C2C2C2C4S3D6D6D6D6C4×S32+ 1+4D4○D12
kernelC429D6C4×D12S3×C22⋊C4Dic35D4C23.26D6C3×C42⋊C2C22×D12C2×D12C42⋊C2C42C22⋊C4C4⋊C4C22×C4C2×C4C6C2
# reps14441111612221824

Matrix representation of C429D6 in GL6(𝔽13)

1200000
0120000
003600
0071000
000036
0000710
,
800000
080000
0010110
0001011
0000120
0000012
,
12120000
100000
00121200
001000
00121211
0010120
,
12120000
010000
0031000
0071000
00310103
0071063

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,0,12,0,0,0,0,11,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,12,1,0,0,12,0,12,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,3,7,3,7,0,0,10,10,10,10,0,0,0,0,10,6,0,0,0,0,3,3] >;

C429D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9D_6
% in TeX

G:=Group("C4^2:9D6");
// GroupNames label

G:=SmallGroup(192,1080);
// by ID

G=gap.SmallGroup(192,1080);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,570,80,6278]);
// Polycyclic

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

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