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G = C426D6order 192 = 26·3

4th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C426D6, C4○D125C4, (C2×D12)⋊12C4, (C4×C12)⋊3C22, C4.83(C2×D12), C424S32C2, C42⋊C24S3, C4.10(D6⋊C4), (C2×Dic6)⋊12C4, D12.22(C2×C4), (C2×C12).144D4, C12.303(C2×D4), (C2×C4).147D12, (C22×C6).78D4, Dic6.23(C2×C4), (C22×C4).129D6, C12.23(C22⋊C4), C12.110(C22×C4), (C2×C12).794C23, C22.25(D6⋊C4), C32(C42⋊C22), C4○D12.38C22, C23.28(C3⋊D4), C4.Dic320C22, (C22×C12).154C22, C4.68(S3×C2×C4), (C2×C4).46(C4×S3), C2.20(C2×D6⋊C4), (C2×C12).94(C2×C4), (C2×C4○D12).8C2, (C2×C6).461(C2×D4), C6.47(C2×C22⋊C4), (C3×C42⋊C2)⋊4C2, (C2×C4).45(C3⋊D4), (C2×C4.Dic3)⋊10C2, C22.27(C2×C3⋊D4), (C2×C6).17(C22⋊C4), (C2×C4).708(C22×S3), SmallGroup(192,564)

Series: Derived Chief Lower central Upper central

C1C12 — C426D6
C1C3C6C2×C6C2×C12C4○D12C2×C4○D12 — C426D6
C3C6C12 — C426D6
C1C4C22×C4C42⋊C2

Generators and relations for C426D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 424 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C3, C4 [×4], C4 [×4], C22 [×3], C22 [×5], S3 [×2], C6, C6 [×3], C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, Dic3 [×2], C12 [×4], C12 [×2], D6 [×4], C2×C6 [×3], C2×C6, C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], C3⋊C8 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×2], C22×S3, C22×C6, C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C4×C12 [×2], C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C42⋊C22, C424S3 [×4], C2×C4.Dic3, C3×C42⋊C2, C2×C4○D12, C426D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C42⋊C22, C2×D6⋊C4, C426D6

Smallest permutation representation of C426D6
On 48 points
Generators in S48
(1 37 16 28)(2 41 17 26)(3 39 18 30)(4 40 13 25)(5 38 14 29)(6 42 15 27)(7 43 22 34)(8 47 23 32)(9 45 24 36)(10 46 19 31)(11 44 20 35)(12 48 21 33)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 46 40 43)(38 47 41 44)(39 48 42 45)
(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 3)(4 6)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 21)(22 24)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J4K6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12N
order12222223444444444446666688881212121212···12
size11222121221122244441212222441212121222224···4

42 irreducible representations

dim1111111122222222244
type+++++++++++
imageC1C2C2C2C2C4C4C4S3D4D4D6D6C4×S3D12C3⋊D4C3⋊D4C42⋊C22C426D6
kernelC426D6C424S3C2×C4.Dic3C3×C42⋊C2C2×C4○D12C2×Dic6C2×D12C4○D12C42⋊C2C2×C12C22×C6C42C22×C4C2×C4C2×C4C2×C4C23C3C1
# reps1411122413121442224

Matrix representation of C426D6 in GL4(𝔽73) generated by

00720
00072
431300
603000
,
665900
14700
006659
00147
,
727200
1000
0011
00720
,
72000
1100
006659
00667
G:=sub<GL(4,GF(73))| [0,0,43,60,0,0,13,30,72,0,0,0,0,72,0,0],[66,14,0,0,59,7,0,0,0,0,66,14,0,0,59,7],[72,1,0,0,72,0,0,0,0,0,1,72,0,0,1,0],[72,1,0,0,0,1,0,0,0,0,66,66,0,0,59,7] >;

C426D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_6D_6
% in TeX

G:=Group("C4^2:6D6");
// GroupNames label

G:=SmallGroup(192,564);
// by ID

G=gap.SmallGroup(192,564);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,1123,1684,102,6278]);
// 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*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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