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G = C2×C3⋊S3⋊3C8  order 288 = 25·32

Direct product of C2 and C3⋊S3⋊3C8

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×C3⋊S3⋊3C8, (C6×C12).9C4, C32⋊3(C22×C8), C62.13(C2×C4), C32⋊2C8⋊10C22, C3⋊Dic3.29C23, (C2×C3⋊S3)⋊7C8, C3⋊S3⋊4(C2×C8), (C3×C6)⋊2(C2×C8), (C4×C3⋊S3).15C4, C4.21(C2×C32⋊C4), (C3×C12).18(C2×C4), C2.1(C22×C32⋊C4), (C2×C32⋊2C8)⋊10C2, (C4×C3⋊S3).95C22, (C22×C3⋊S3).14C4, (C2×C4).11(C32⋊C4), C3⋊Dic3.50(C2×C4), (C3×C6).24(C22×C4), C22.15(C2×C32⋊C4), (C2×C3⋊Dic3).172C22, (C2×C4×C3⋊S3).25C2, (C2×C3⋊S3).44(C2×C4), SmallGroup(288,929)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C3⋊S3⋊3C8
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8 — C2×C3⋊S3⋊3C8
Lower central
C32 — C2×C3⋊S3⋊3C8
Upper central
C1 — C2×C4

Generators and relations for C2×C3⋊S3⋊3C8
 G = < a,b,c,d,e | a2=b3=c3=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc-1, dcd=c-1, ece-1=b-1c-1, de=ed >

Subgroups: 576 in 130 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C4 [×2], C4 [×2], C22, C22 [×6], S3 [×8], C6 [×6], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C2×C8 [×6], C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C22×C8, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×2], C32⋊2C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C3⋊S3⋊3C8 [×4], C2×C32⋊2C8 [×2], C2×C4×C3⋊S3, C2×C3⋊S3⋊3C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, C22×C8, C32⋊C4, C2×C32⋊C4 [×3], C3⋊S3⋊3C8 [×2], C22×C32⋊C4, C2×C3⋊S3⋊3C8

Smallest permutation representation of C2×C3⋊S3⋊3C8
►On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)

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

(2 19 29)(4 31 21)(6 23 25)(8 27 17)(10 44 36)(12 38 46)(14 48 40)(16 34 42)

(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H6A···6F8A···8P12A···12H
order1222222233444444446···68···812···12
size1111999944111199994···49···94···4

48 irreducible representations

dim111111114444
type+++++++
imageC1C2C2C2C4C4C4C8C32⋊C4C2×C32⋊C4C2×C32⋊C4C3⋊S3⋊3C8
kernelC2×C3⋊S3⋊3C8C3⋊S3⋊3C8C2×C32⋊2C8C2×C4×C3⋊S3C4×C3⋊S3C6×C12C22×C3⋊S3C2×C3⋊S3C2×C4C4C22C2
# reps1421422162428

Matrix representation of C2×C3⋊S3⋊3C8 ►in GL5(𝔽73)

720000
072000
007200
000720
000072
,
10000
007200
017200
000721
000720
,
10000
01000
00100
000072
000172
,
720000
00100
01000
00001
00010
,
720000
00010
00001
002700
027000

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[72,0,0,0,0,0,0,0,0,27,0,0,0,27,0,0,1,0,0,0,0,0,1,0,0] >;
 

C2×C3⋊S3⋊3C8 in GAP, Magma, Sage, TeX 

C_2\times C_3\rtimes S_3\rtimes_3C_8
 
% in TeX
 
G:=Group("C2xC3:S3:3C8");
 
// GroupNames label
 
G:=SmallGroup(288,929);
 
// by ID
 
G=gap.SmallGroup(288,929);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,100,80,9413,362,12550,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1,e*b*e^-1=b*c^-1,d*c*d=c^-1,e*c*e^-1=b^-1*c^-1,d*e=e*d>;
 
// generators/relations
 

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