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G = C4210D6order 192 = 26·3

8th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4210D6, C4⋊C442D6, (C4×D12)⋊7C2, (C2×C12)⋊10D4, (C2×C4)⋊11D12, D61(C4○D4), (C4×C12)⋊5C22, C4.70(C2×D12), C12⋊D442C2, D6⋊D429C2, C42⋊C28S3, D6⋊C451C22, C4.D1247C2, C12.223(C2×D4), (C2×C6).68C24, C22⋊C4.92D6, C6.12(C22×D4), (C2×D12)⋊52C22, C4⋊Dic355C22, (C22×C4).381D6, C2.14(C22×D12), C22.19(C2×D12), (C2×C12).143C23, C31(C22.19C24), (C2×Dic6)⋊61C22, C22.97(S3×C23), C23.21D632C2, (C22×S3).18C23, C23.166(C22×S3), (C22×C6).138C23, (S3×C23).104C22, (C22×C12).228C22, (C2×Dic3).197C23, (C22×Dic3).217C22, (S3×C22×C4)⋊2C2, C2.9(S3×C4○D4), (S3×C2×C4)⋊44C22, (C2×C6).49(C2×D4), (C2×C4○D12)⋊17C2, (C3×C4⋊C4)⋊52C22, C6.133(C2×C4○D4), (C3×C42⋊C2)⋊10C2, (C2×C4).575(C22×S3), (C2×C3⋊D4).99C22, (C3×C22⋊C4).100C22, SmallGroup(192,1083)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4210D6
C1C3C6C2×C6C22×S3S3×C23S3×C22×C4 — C4210D6
C3C2×C6 — C4210D6
C1C2×C4C42⋊C2

Generators and relations for C4210D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 904 in 330 conjugacy classes, 115 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], S3 [×6], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], D4 [×14], Q8 [×2], C23, C23 [×10], Dic3 [×4], C12 [×4], C12 [×4], D6 [×4], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic6 [×2], C4×S3 [×12], D12 [×10], C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×4], C22×S3 [×6], C22×C6, C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4⋊Dic3 [×4], D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×6], S3×C2×C4 [×4], C2×D12, C2×D12 [×4], C4○D12 [×4], C22×Dic3, C2×C3⋊D4 [×2], C22×C12, S3×C23, C22.19C24, C4×D12 [×4], D6⋊D4 [×2], C23.21D6 [×2], C12⋊D4 [×2], C4.D12 [×2], C3×C42⋊C2, S3×C22×C4, C2×C4○D12, C4210D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C2×D12 [×6], S3×C23, C22.19C24, C22×D12, S3×C4○D4 [×2], C4210D6

Smallest permutation representation of C4210D6
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 14)(2 13)(3 18)(4 17)(5 16)(6 15)(7 43)(8 48)(9 47)(10 46)(11 45)(12 44)(19 27)(20 26)(21 25)(22 30)(23 29)(24 28)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P6A6B6C6D6E12A12B12C12D12E···12N
order12222222222234444444444444444666661212121212···12
size1111226666121221111224444666612122224422224···4

48 irreducible representations

dim111111111222222224
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4○D4D12S3×C4○D4
kernelC4210D6C4×D12D6⋊D4C23.21D6C12⋊D4C4.D12C3×C42⋊C2S3×C22×C4C2×C4○D12C42⋊C2C2×C12C42C22⋊C4C4⋊C4C22×C4D6C2×C4C2
# reps142222111142221884

Matrix representation of C4210D6 in GL6(𝔽13)

0120000
100000
000100
0012000
0000120
0000012
,
100000
010000
008000
000800
000010
000001
,
1200000
0120000
0012000
000100
0000012
0000112
,
1200000
010000
001000
0001200
0000112
0000012

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

C4210D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{10}D_6
% in TeX

G:=Group("C4^2:10D6");
// GroupNames label

G:=SmallGroup(192,1083);
// by ID

G=gap.SmallGroup(192,1083);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,80,6278]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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