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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, (C22×C6).78C23, C23.80(C22×S3), C3⋊5(C22.54C24), C6.D4⋊38C22, C23.23D6⋊28C2, (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

Subgroups: 768 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×9], C22, C22 [×22], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×9], D4 [×12], C23, C23 [×3], C23 [×5], Dic3 [×6], C12 [×3], D6 [×10], C2×C6, C2×C6 [×12], C42, C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×6], C2×D4 [×6], C24, C2×Dic3 [×6], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×6], C22×S3 [×2], C22×S3 [×3], C22×C6, C22×C6 [×3], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2 [×2], C4⋊1D4, Dic3⋊C4 [×6], D6⋊C4 [×6], C6.D4 [×6], C4×C12, C22×Dic3 [×3], C2×C3⋊D4 [×6], C6×D4 [×6], S3×C23, C22.54C24, C42⋊3S3 [×2], C23.23D6 [×3], C23⋊2D6 [×3], C23.14D6 [×6], C3×C4⋊1D4, C42⋊30D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4) [×3], S3×C23, C22.54C24, D4⋊6D6 [×3], C42⋊30D6

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

33 conjugacy classes

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

33 irreducible representations

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

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
 

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