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G = C42⋊11D6  order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊11D6, C6.952+ 1+4, C4⋊C4⋊43D6, (C2×C4)⋊5D12, (C2×C12)⋊11D4, D6⋊D4⋊4C2, C4⋊D12⋊3C2, (C4×C12)⋊1C22, D6⋊C4⋊3C22, C4.71(C2×D12), C12⋊D4⋊11C2, C42⋊C2⋊9S3, C42⋊7S3⋊3C2, 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, C3⋊1(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

Derived series
C1 — C2×C6 — C42⋊11D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C42⋊11D6
Lower central
C3 — C2×C6 — C42⋊11D6
Upper central
C1 — C22 — C42⋊C2

Generators and relations for C42⋊11D6
 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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, S3×C23, C22.29C24, C4⋊D12, C42⋊7S3, D6⋊D4, C12⋊D4, C3×C42⋊C2, C22×D12, C2×C4○D12, C42⋊11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, 2+ 1+4, C2×D12, S3×C23, C22.29C24, C22×D12, D4○D12, C42⋊11D6

Smallest permutation representation of C42⋊11D6
►On 48 points
Generators in S48
(1 26 4 19)(2 23 5 30)(3 28 6 21)(7 32 44 35)(8 40 45 37)(9 34 46 31)(10 42 47 39)(11 36 48 33)(12 38 43 41)(13 22 16 29)(14 27 17 20)(15 24 18 25)

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

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

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
kernelC42⋊11D6C4⋊D12C42⋊7S3D6⋊D4C12⋊D4C3×C42⋊C2C22×D12C2×C4○D12C42⋊C2C2×C12C42C22⋊C4C4⋊C4C22×C4C2×C4C6C2
# reps12244111142221824

Matrix representation of C42⋊11D6 ►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] >;
 

C42⋊11D6 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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