Copied to
clipboard

G = C42⋊30D6  order 192 = 26·3

28th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊30D6, C6.802+ 1+4, (C2×D4)⋊13D6, C4⋊1D4⋊10S3, C23⋊2D6⋊28C2, (C4×C12)⋊37C22, (C6×D4)⋊34C22, C42⋊3S3⋊19C2, (C2×C6).264C24, D6⋊C4.75C22, C23.14D6⋊39C2, C2.84(D4⋊6D6), (C2×C12).638C23, Dic3⋊C4⋊37C22, C23.80(C22×S3), (C22×C6).78C23, C3⋊5(C22.54C24), C23.23D6⋊28C2, C6.D4⋊38C22, (S3×C23).73C22, C22.285(S3×C23), (C22×S3).118C23, (C2×Dic3).138C23, (C22×Dic3)⋊30C22, (C3×C4⋊1D4)⋊15C2, (C2×C4).216(C22×S3), (C2×C3⋊D4).80C22, SmallGroup(192,1279)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C42⋊30D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C23⋊2D6 — C42⋊30D6
Lower central
C3 — C2×C6 — C42⋊30D6
Upper central
C1 — C22 — C4⋊1D4

Generators and relations for C42⋊30D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >

Subgroups: 768 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, Dic3⋊C4, D6⋊C4, C6.D4, C4×C12, C22×Dic3, C2×C3⋊D4, C6×D4, S3×C23, C22.54C24, C42⋊3S3, C23.23D6, C23⋊2D6, C23.14D6, C3×C4⋊1D4, C42⋊30D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, C22.54C24, D4⋊6D6, C42⋊30D6

Smallest permutation representation of C42⋊30D6
►On 48 points
Generators in S48
(1 16 7 13)(2 14 8 17)(3 18 9 15)(4 36 12 33)(5 34 10 31)(6 32 11 35)(19 38 27 48)(20 43 28 39)(21 40 29 44)(22 45 30 41)(23 42 25 46)(24 47 26 37)

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D···4I6A6B6C6D6E6F6G12A···12F
order122222222234444···4666666612···12
size111144441212244412···1222288884···4

33 irreducible representations

dim11111122244
type++++++++++
imageC1C2C2C2C2C2S3D6D62+ 1+4D4⋊6D6
kernelC42⋊30D6C42⋊3S3C23.23D6C23⋊2D6C23.14D6C3×C4⋊1D4C4⋊1D4C42C2×D4C6C2
# reps12336111636

Matrix representation of C42⋊30D6 ►in GL8(𝔽13)

00100000
000120000
120000000
01000000
000000120
000000012
00001000
00000100
,
01000000
120000000
000120000
00100000
00002900
000041100
00000029
000000411
,
120000000
01000000
00100000
000120000
000001200
000011200
00000001
000000121
,
120000000
01000000
001200000
00010000
000011200
000001200
000000121
00000001

G:=sub<GL(8,GF(13))| [0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,9,11,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,9,11],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1] >;
 

C42⋊30D6 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{30}D_6
 
% in TeX
 
G:=Group("C4^2:30D6");
 
// GroupNames label
 
G:=SmallGroup(192,1279);
 
// by ID
 
G=gap.SmallGroup(192,1279);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,1571,570,297,136,6278]);
 
// 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^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁