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G = C4211D6order 192 = 26·3

9th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4211D6, C6.952+ 1+4, C4⋊C443D6, (C2×C4)⋊5D12, (C2×C12)⋊11D4, D6⋊D44C2, C4⋊D123C2, (C4×C12)⋊1C22, D6⋊C43C22, C4.71(C2×D12), C12⋊D411C2, C42⋊C29S3, C427S33C2, C2.7(D4○D12), C12.287(C2×D4), (C2×D12)⋊5C22, (C2×C6).69C24, C22⋊C4.93D6, C6.13(C22×D4), (C22×D12)⋊14C2, C2.15(C22×D12), (C22×C4).206D6, C22.20(C2×D12), (C2×C12).144C23, C31(C22.29C24), (C2×Dic6)⋊51C22, C22.98(S3×C23), (C22×S3).19C23, (S3×C23).36C22, (C22×C6).139C23, C23.167(C22×S3), (C2×Dic3).23C23, (C22×C12).229C22, (S3×C2×C4)⋊1C22, (C2×C6).50(C2×D4), (C2×C4○D12)⋊18C2, (C3×C4⋊C4)⋊53C22, (C3×C42⋊C2)⋊11C2, (C2×C4).149(C22×S3), (C2×C3⋊D4).100C22, (C3×C22⋊C4).101C22, SmallGroup(192,1084)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4211D6
C1C3C6C2×C6C22×S3S3×C23C22×D12 — C4211D6
C3C2×C6 — C4211D6
C1C22C42⋊C2

Generators and relations for C4211D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1128 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], S3 [×6], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×22], Q8 [×2], C23, C23 [×14], Dic3 [×2], C12 [×4], C12 [×4], D6 [×26], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×19], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×4], D12 [×18], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×6], C22×S3 [×8], C22×C6, C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C2×D12 [×12], C2×D12 [×4], C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, S3×C23 [×2], C22.29C24, C4⋊D12 [×2], C427S3 [×2], D6⋊D4 [×4], C12⋊D4 [×4], C3×C42⋊C2, C22×D12, C2×C4○D12, C4211D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×D12 [×6], S3×C23, C22.29C24, C22×D12, D4○D12 [×2], C4211D6

Smallest permutation representation of C4211D6
On 48 points
Generators in S48
(1 26 4 21)(2 19 5 30)(3 28 6 23)(7 32 46 35)(8 40 47 37)(9 34 48 31)(10 42 43 39)(11 36 44 33)(12 38 45 41)(13 22 16 27)(14 25 17 20)(15 24 18 29)
(1 34 15 41)(2 35 16 42)(3 36 17 37)(4 31 18 38)(5 32 13 39)(6 33 14 40)(7 27 43 19)(8 28 44 20)(9 29 45 21)(10 30 46 22)(11 25 47 23)(12 26 48 24)
(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 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 35)(14 34)(15 33)(16 32)(17 31)(18 36)(19 43)(20 48)(21 47)(22 46)(23 45)(24 44)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F···2K 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E12A12B12C12D12E···12N
order1222222···234444444444666661212121212···12
size11112212···1222222444412122224422224···4

42 irreducible representations

dim11111111222222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D6D6D6D6D122+ 1+4D4○D12
kernelC4211D6C4⋊D12C427S3D6⋊D4C12⋊D4C3×C42⋊C2C22×D12C2×C4○D12C42⋊C2C2×C12C42C22⋊C4C4⋊C4C22×C4C2×C4C6C2
# reps12244111142221824

Matrix representation of C4211D6 in GL6(𝔽13)

360000
7100000
0010110
0001011
0010120
0001012
,
1200000
0120000
0010600
007300
0000106
000073
,
12120000
100000
000100
0012100
0001012
00121112
,
110000
0120000
003300
0061000
00331010
0061073

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

C4211D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{11}D_6
% in TeX

G:=Group("C4^2:11D6");
// GroupNames label

G:=SmallGroup(192,1084);
// by ID

G=gap.SmallGroup(192,1084);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,80,6278]);
// Polycyclic

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

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