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G = C4213D6order 192 = 26·3

11st semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4213D6, C6.152+ 1+4, C4⋊C456D6, D49(C4×S3), (C4×D4)⋊7S3, (S3×D4)⋊5C4, (D4×C12)⋊9C2, (C22×C4)⋊7D6, (C4×D12)⋊24C2, D1213(C2×C4), C22⋊C453D6, (D4×Dic3)⋊8C2, (C4×C12)⋊16C22, D6⋊C462C22, (C2×D4).245D6, C2.3(D4○D12), (C2×C6).89C24, C6.22(C23×C4), Dic35D414C2, D6.8(C22×C4), C422S311C2, C2.3(D46D6), C12.32(C22×C4), C4⋊Dic373C22, Dic34D445C2, (C2×C12).587C23, Dic3⋊C464C22, (C22×C12)⋊36C22, C33(C22.11C24), (C4×Dic3)⋊11C22, (C6×D4).253C22, C22.32(S3×C23), Dic3.9(C22×C4), (C2×D12).258C22, C6.D448C22, (S3×C23).38C22, C23.178(C22×S3), (C22×C6).159C23, (C22×Dic3)⋊8C22, (C22×S3).168C23, (C2×Dic3).201C23, C4.32(S3×C2×C4), (C2×S3×D4).7C2, (C4×S3)⋊3(C2×C4), C3⋊D43(C2×C4), (C2×D6⋊C4)⋊34C2, C22.2(S3×C2×C4), (C3×D4)⋊12(C2×C4), (C4×C3⋊D4)⋊40C2, (S3×C2×C4)⋊46C22, C4⋊C47S314C2, C2.24(S3×C22×C4), (S3×C22⋊C4)⋊27C2, (C3×C4⋊C4)⋊56C22, (C22×S3)⋊8(C2×C4), (C2×C6).2(C22×C4), (C3×C22⋊C4)⋊63C22, (C2×C4).282(C22×S3), (C2×C3⋊D4).110C22, SmallGroup(192,1104)

Series: Derived Chief Lower central Upper central

C1C6 — C4213D6
C1C3C6C2×C6C22×S3S3×C23C2×S3×D4 — C4213D6
C3C6 — C4213D6
C1C22C4×D4

Generators and relations for C4213D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 904 in 338 conjugacy classes, 151 normal (43 characteristic)
C1, C2 [×3], C2 [×10], C3, C4 [×2], C4 [×10], C22, C22 [×4], C22 [×24], S3 [×6], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×12], C23 [×2], C23 [×15], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], D6 [×14], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×11], C24 [×2], C4×S3 [×4], C4×S3 [×4], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3, C22×S3 [×10], C22×S3 [×4], C22×C6 [×2], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4, C4×D4 [×7], C22×D4, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×4], C2×D12, S3×D4 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, S3×C23 [×2], C22.11C24, C422S3, C4×D12, S3×C22⋊C4 [×2], Dic34D4 [×2], C4⋊C47S3, Dic35D4, C2×D6⋊C4 [×2], C4×C3⋊D4 [×2], D4×Dic3, D4×C12, C2×S3×D4, C4213D6
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ 1+4 [×2], S3×C2×C4 [×6], S3×C23, C22.11C24, S3×C22×C4, D46D6, D4○D12, C4213D6

Smallest permutation representation of C4213D6
On 48 points
Generators in S48
(1 41 17 31)(2 32 18 42)(3 37 13 33)(4 34 14 38)(5 39 15 35)(6 36 16 40)(7 23 44 27)(8 28 45 24)(9 19 46 29)(10 30 47 20)(11 21 48 25)(12 26 43 22)
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 42 47 35)(8 37 48 36)(9 38 43 31)(10 39 44 32)(11 40 45 33)(12 41 46 34)(13 25 16 28)(14 26 17 29)(15 27 18 30)
(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 47)(8 46)(9 45)(10 44)(11 43)(12 48)(13 14)(15 18)(16 17)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2M 3 4A···4J4K···4T6A6B6C6D6E6F6G12A12B12C12D12E···12L
order122222222···234···44···466666661212121212···12
size111122226···622···26···6222444422224···4

54 irreducible representations

dim11111111111112222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4S3D6D6D6D6D6C4×S32+ 1+4D46D6D4○D12
kernelC4213D6C422S3C4×D12S3×C22⋊C4Dic34D4C4⋊C47S3Dic35D4C2×D6⋊C4C4×C3⋊D4D4×Dic3D4×C12C2×S3×D4S3×D4C4×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D4C6C2C2
# reps111221122111161121218222

Matrix representation of C4213D6 in GL6(𝔽13)

1200000
0120000
0000120
0000012
001000
000100
,
500000
050000
000100
0012000
000001
0000120
,
0120000
1120000
0012000
0001200
000010
000001
,
1210000
010000
0012000
000100
000010
0000012

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;

C4213D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{13}D_6
% in TeX

G:=Group("C4^2:13D6");
// GroupNames label

G:=SmallGroup(192,1104);
// by ID

G=gap.SmallGroup(192,1104);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,80,6278]);
// Polycyclic

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

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