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G = C24⋊9D6  order 192 = 26·3

4th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24⋊9D6, C6.302+ (1+4), C22⋊C4⋊8D6, C22≀C2⋊8S3, C23⋊2D6⋊7C2, (C2×D4).87D6, C24⋊4S3⋊9C2, D6⋊C4⋊15C22, (C2×C12).32C23, (C2×C6).138C24, (S3×C23)⋊8C22, (C23×C6)⋊11C22, C2.32(D4⋊6D6), C23.12D6⋊13C2, C3⋊1(C24⋊C22), (C4×Dic3)⋊18C22, (C2×Dic6)⋊23C22, (C6×D4).112C22, C23.11D6⋊15C2, C6.D4⋊18C22, (C22×S3).57C23, C22.159(S3×C23), C23.120(C22×S3), (C22×C6).183C23, (C2×Dic3).63C23, (C3×C22≀C2)⋊9C2, (C3×C22⋊C4)⋊8C22, (C2×C4).32(C22×S3), (C2×C3⋊D4).22C22, SmallGroup(192,1153)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24⋊9D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C23⋊2D6 — C24⋊9D6
Lower central
C3 — C2×C6 — C24⋊9D6

Subgroups: 800 in 260 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×9], C22, C22 [×26], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×6], D4 [×9], Q8 [×3], C23, C23 [×3], C23 [×8], Dic3 [×6], C12 [×3], D6 [×10], C2×C6, C2×C6 [×16], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×15], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×3], C24, C24, Dic6 [×3], C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C22×S3 [×2], C22×S3 [×3], C22×C6, C22×C6 [×3], C22×C6 [×3], C22≀C2, C22≀C2 [×5], C4.4D4 [×9], C4×Dic3 [×3], D6⋊C4 [×6], C6.D4 [×9], C3×C22⋊C4 [×3], C2×Dic6 [×3], C2×C3⋊D4 [×6], C6×D4 [×3], S3×C23, C23×C6, C24⋊C22, C23.11D6 [×6], C23.12D6 [×3], C23⋊2D6 [×3], C24⋊4S3 [×2], C3×C22≀C2, C24⋊9D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4) [×3], S3×C23, C24⋊C22, D4⋊6D6 [×3], C24⋊9D6

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=acd, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽13)

00100000
00010000
10000000
01000000
00000010
000011021
00001000
000011120
,
119000000
42000000
001190000
00420000
00001300
000001200
00007073
000036106
,
120000000
012000000
001200000
000120000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
1212000000
10000000
00110000
001200000
00001000
000081200
00000010
000040412
,
1212000000
01000000
00110000
000120000
000012000
000001200
00000010
00009001

G:=sub<GL(8,GF(13))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,1,11,0,0,0,0,0,0,0,1,0,0,0,0,1,2,0,2,0,0,0,0,0,1,0,0],[11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,1,0,7,3,0,0,0,0,3,12,0,6,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,8,0,4,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,9,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D···4I6A6B6C6D···6I6J12A12B12C
order122222222234444···46666···66121212
size111144441212244412···122224···48888

33 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2S3D6D6D62+ (1+4)D4⋊6D6
kernelC24⋊9D6C23.11D6C23.12D6C23⋊2D6C24⋊4S3C3×C22≀C2C22≀C2C22⋊C4C2×D4C24C6C2
# reps163321133136

In GAP, Magma, Sage, TeX 

C_2^4\rtimes_9D_6
 
% in TeX
 
G:=Group("C2^4:9D6");
 
// GroupNames label
 
G:=SmallGroup(192,1153);
 
// by ID
 
G=gap.SmallGroup(192,1153);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,1571,570,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,f*b*f=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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