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G = C23⋊3Dic6  order 192 = 26·3

2nd semidirect product of C23 and Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊3Dic6, C24.34D6, C6.12+ (1+4), (C22×C6)⋊5Q8, C6.6(C22×Q8), (C2×C6).27C24, C4⋊Dic3⋊3C22, C22⋊C4.86D6, C3⋊1(C23⋊2Q8), C12.48D4⋊3C2, C2.6(D4⋊6D6), Dic3⋊C4⋊1C22, (C2×Dic6)⋊2C22, (C22×C4).185D6, C2.8(C22×Dic6), C22.5(C2×Dic6), (C2×C12).127C23, Dic3.D4⋊1C2, C22.69(S3×C23), (C23×C6).53C22, (C2×Dic3).8C23, (C22×C12).71C22, C23.154(C22×S3), (C22×C6).119C23, C6.D4.85C22, (C22×Dic3).76C22, (C2×C6).49(C2×Q8), (C2×C22⋊C4).18S3, (C6×C22⋊C4).18C2, (C2×C4).133(C22×S3), (C2×C6.D4).22C2, (C3×C22⋊C4).97C22, SmallGroup(192,1042)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C23⋊3Dic6
Chief series
C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C6.D4 — C23⋊3Dic6
Lower central
C3 — C2×C6 — C23⋊3Dic6

Subgroups: 584 in 242 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×12], C22, C22 [×6], C22 [×10], C6, C6 [×2], C6 [×6], C2×C4 [×4], C2×C4 [×14], Q8 [×4], C23, C23 [×6], C23 [×2], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C2×C6 [×10], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C24, Dic6 [×4], C2×Dic3 [×8], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×2], C22×C6, C22×C6 [×6], C22×C6 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C22⋊Q8 [×12], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×8], C3×C22⋊C4 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23⋊2Q8, Dic3.D4 [×8], C12.48D4 [×4], C2×C6.D4 [×2], C6×C22⋊C4, C23⋊3Dic6

Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, 2+ (1+4) [×2], C2×Dic6 [×6], S3×C23, C23⋊2Q8, C22×Dic6, D4⋊6D6 [×2], C23⋊3Dic6

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=d6, ab=ba, dad-1=ac=ca, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Smallest permutation representation
►On 48 points
Generators in S48
(2 31)(4 33)(6 35)(8 25)(10 27)(12 29)(13 46)(15 48)(17 38)(19 40)(21 42)(23 44)

(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)

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

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

Matrix representation ►G ⊆ GL8(𝔽13)

120000000
012000000
001200000
000120000
00001000
000001200
00000010
000000012
,
10000000
01000000
00100000
00010000
00001000
00000100
000000120
000000012
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
012000000
10000000
001210000
001200000
00000100
000012000
000000012
00000010
,
80000000
05000000
001200000
001210000
00000010
00000001
000012000
000001200

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

42 conjugacy classes

class 1 2A2B2C2D···2I 3 4A4B4C4D4E···4L6A···6G6H6I6J6K12A···12H
order12222···2344444···46···6666612···12
size11112···22444412···122···244444···4

42 irreducible representations

dim1111122222244
type++++++-+++-+
imageC1C2C2C2C2S3Q8D6D6D6Dic62+ (1+4)D4⋊6D6
kernelC23⋊3Dic6Dic3.D4C12.48D4C2×C6.D4C6×C22⋊C4C2×C22⋊C4C22×C6C22⋊C4C22×C4C24C23C6C2
# reps1842114421824

In GAP, Magma, Sage, TeX 

C_2^3\rtimes_3Dic_6
 
% in TeX
 
G:=Group("C2^3:3Dic6");
 
// GroupNames label
 
G:=SmallGroup(192,1042);
 
// by ID
 
G=gap.SmallGroup(192,1042);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,758,675,570,80,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=1,e^2=d^6,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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