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G = C2×C3⋊S33C8order 288 = 25·32

Direct product of C2 and C3⋊S33C8

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

Aliases: C2×C3⋊S33C8, (C6×C12).9C4, C323(C22×C8), C62.13(C2×C4), C322C810C22, C3⋊Dic3.29C23, (C2×C3⋊S3)⋊7C8, C3⋊S34(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×C322C8)⋊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

C1C32 — C2×C3⋊S33C8
C1C32C3×C6C3⋊Dic3C322C8C2×C322C8 — C2×C3⋊S33C8
C32 — C2×C3⋊S33C8
C1C2×C4

Generators and relations for C2×C3⋊S33C8
 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], C322C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C3⋊S33C8 [×4], C2×C322C8 [×2], C2×C4×C3⋊S3, C2×C3⋊S33C8
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⋊S33C8 [×2], C22×C32⋊C4, C2×C3⋊S33C8

Smallest permutation representation of C2×C3⋊S33C8
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⋊S33C8
kernelC2×C3⋊S33C8C3⋊S33C8C2×C322C8C2×C4×C3⋊S3C4×C3⋊S3C6×C12C22×C3⋊S3C2×C3⋊S3C2×C4C4C22C2
# reps1421422162428

Matrix representation of C2×C3⋊S33C8 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⋊S33C8 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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