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G = C42⋊13D6  order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊13D6, C6.152+ 1+4, C4⋊C4⋊56D6, D4⋊9(C4×S3), (C4×D4)⋊7S3, (S3×D4)⋊5C4, (D4×C12)⋊9C2, (C22×C4)⋊7D6, (C4×D12)⋊24C2, D12⋊13(C2×C4), C22⋊C4⋊53D6, (D4×Dic3)⋊8C2, (C4×C12)⋊16C22, D6⋊C4⋊62C22, (C2×D4).245D6, C2.3(D4○D12), (C2×C6).89C24, C6.22(C23×C4), Dic3⋊5D4⋊14C2, D6.8(C22×C4), C42⋊2S3⋊11C2, C2.3(D4⋊6D6), C12.32(C22×C4), C4⋊Dic3⋊73C22, Dic3⋊4D4⋊45C2, (C2×C12).587C23, Dic3⋊C4⋊64C22, (C22×C12)⋊36C22, C3⋊3(C22.11C24), (C4×Dic3)⋊11C22, (C6×D4).253C22, C22.32(S3×C23), Dic3.9(C22×C4), (C2×D12).258C22, C6.D4⋊48C22, (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⋊D4⋊3(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⋊C4⋊7S3⋊14C2, 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

Derived series
C1 — C6 — C42⋊13D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C42⋊13D6
Lower central
C3 — C6 — C42⋊13D6
Upper central
C1 — C22 — C4×D4

Generators and relations for C42⋊13D6
 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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D4, C22×D4, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C22.11C24, C42⋊2S3, C4×D12, S3×C22⋊C4, Dic3⋊4D4, C4⋊C4⋊7S3, Dic3⋊5D4, C2×D6⋊C4, C4×C3⋊D4, D4×Dic3, D4×C12, C2×S3×D4, C42⋊13D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, 2+ 1+4, S3×C2×C4, S3×C23, C22.11C24, S3×C22×C4, D4⋊6D6, D4○D12, C42⋊13D6

Smallest permutation representation of C42⋊13D6
►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 19 44 27)(8 28 45 20)(9 21 46 29)(10 30 47 22)(11 23 48 25)(12 26 43 24)

(1 21 4 24)(2 22 5 19)(3 23 6 20)(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 30)(20 29)(21 28)(22 27)(23 26)(24 25)(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)(7,19,44,27)(8,28,45,20)(9,21,46,29)(10,30,47,22)(11,23,48,25)(12,26,43,24), (1,21,4,24)(2,22,5,19)(3,23,6,20)(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)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(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)(7,19,44,27)(8,28,45,20)(9,21,46,29)(10,30,47,22)(11,23,48,25)(12,26,43,24), (1,21,4,24)(2,22,5,19)(3,23,6,20)(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)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(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),(7,19,44,27),(8,28,45,20),(9,21,46,29),(10,30,47,22),(11,23,48,25),(12,26,43,24)], [(1,21,4,24),(2,22,5,19),(3,23,6,20),(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),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(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+4D4⋊6D6D4○D12
kernelC42⋊13D6C42⋊2S3C4×D12S3×C22⋊C4Dic3⋊4D4C4⋊C4⋊7S3Dic3⋊5D4C2×D6⋊C4C4×C3⋊D4D4×Dic3D4×C12C2×S3×D4S3×D4C4×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D4C6C2C2
# reps111221122111161121218222

Matrix representation of C42⋊13D6 ►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] >;
 

C42⋊13D6 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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