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G = C4⋊C419D6order 192 = 26·3

2nd semidirect product of C4⋊C4 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C419D6, (S3×D4)⋊2C4, (C2×C8)⋊19D6, (C4×S3).3D4, D4.9(C4×S3), D12.1(C2×C4), C6.D85C2, C4.155(S3×D4), D4⋊C416S3, (C2×C24)⋊28C22, (C2×D4).131D6, D4⋊Dic34C2, C2.D2423C2, C12.104(C2×D4), C2.2(D8⋊S3), C2.1(Q83D6), C12.5(C22×C4), C22.69(S3×D4), C6.29(C8⋊C22), C4⋊Dic318C22, (C6×D4).31C22, (C22×S3).68D4, (C2×C12).210C23, D6.11(C22⋊C4), (C2×Dic3).139D4, (C2×D12).48C22, C31(C23.37D4), Dic3.7(C22⋊C4), C4.5(S3×C2×C4), (C2×S3×D4).4C2, (C2×C3⋊C8)⋊1C22, C4⋊C47S31C2, (C3×C4⋊C4)⋊2C22, (C4×S3).1(C2×C4), (C3×D4).3(C2×C4), (C2×C8⋊S3)⋊15C2, (S3×C2×C4).5C22, (C2×C6).223(C2×D4), C2.18(S3×C22⋊C4), C6.17(C2×C22⋊C4), (C3×D4⋊C4)⋊24C2, (C2×C4).317(C22×S3), SmallGroup(192,329)

Series: Derived Chief Lower central Upper central

C1C12 — C4⋊C419D6
C1C3C6C2×C6C2×C12S3×C2×C4C2×S3×D4 — C4⋊C419D6
C3C6C12 — C4⋊C419D6
C1C22C2×C4D4⋊C4

Generators and relations for C4⋊C419D6
 G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab-1, dcd=c-1 >

Subgroups: 680 in 190 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×20], S3 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4 [×8], C23 [×11], Dic3 [×2], Dic3, C12 [×2], C12, D6 [×2], D6 [×14], C2×C6, C2×C6 [×4], C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2) [×2], C22×C4, C2×D4, C2×D4 [×8], C24, C3⋊C8, C24, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C22×S3, C22×S3 [×9], C22×C6, D4⋊C4, D4⋊C4 [×3], C42⋊C2, C2×M4(2), C22×D4, C8⋊S3 [×2], C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, S3×D4 [×4], S3×D4 [×2], C2×C3⋊D4, C6×D4, S3×C23, C23.37D4, C6.D8, C2.D24, D4⋊Dic3, C3×D4⋊C4, C4⋊C47S3, C2×C8⋊S3, C2×S3×D4, C4⋊C419D6
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], C22×S3, C2×C22⋊C4, C8⋊C22 [×2], S3×C2×C4, S3×D4 [×2], C23.37D4, S3×C22⋊C4, D8⋊S3, Q83D6, C4⋊C419D6

Smallest permutation representation of C4⋊C419D6
On 48 points
Generators in S48
(1 32 19 8)(2 9 20 33)(3 34 21 10)(4 11 22 35)(5 36 23 12)(6 7 24 31)(13 16 38 41)(14 42 39 17)(15 18 40 37)(25 28 48 45)(26 46 43 29)(27 30 44 47)
(1 29 4 40)(2 38 5 27)(3 25 6 42)(7 14 34 45)(8 26 35 37)(9 16 36 47)(10 28 31 39)(11 18 32 43)(12 30 33 41)(13 23 44 20)(15 19 46 22)(17 21 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 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 41)(14 40)(15 39)(16 38)(17 37)(18 42)(19 24)(20 23)(21 22)(25 43)(26 48)(27 47)(28 46)(29 45)(30 44)(31 32)(33 36)(34 35)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
order12222222223444444446666688881212121224242424
size111144661212222446612122228844121244884444

36 irreducible representations

dim1111111112222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4S3D4D4D4D6D6D6C4×S3C8⋊C22S3×D4S3×D4D8⋊S3Q83D6
kernelC4⋊C419D6C6.D8C2.D24D4⋊Dic3C3×D4⋊C4C4⋊C47S3C2×C8⋊S3C2×S3×D4S3×D4D4⋊C4C4×S3C2×Dic3C22×S3C4⋊C4C2×C8C2×D4D4C6C4C22C2C2
# reps1111111181211111421122

Matrix representation of C4⋊C419D6 in GL6(𝔽73)

7200000
0720000
0007200
001000
000001
0000720
,
4600000
0460000
000010
000001
0072000
0007200
,
0720000
1720000
0072000
000100
0000072
0000720
,
1720000
0720000
0072000
000100
000001
000010

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C4⋊C419D6 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_{19}D_6
% in TeX

G:=Group("C4:C4:19D6");
// GroupNames label

G:=SmallGroup(192,329);
// by ID

G=gap.SmallGroup(192,329);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,58,851,438,102,6278]);
// Polycyclic

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

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