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## G = C2×C32⋊D8order 288 = 25·32

### Direct product of C2 and C32⋊D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C2×C32⋊D8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C2×C32⋊D8
 Lower central C32 — C3×C6 — C3⋊Dic3 — C2×C32⋊D8
 Upper central C1 — C22

Generators and relations for C2×C32⋊D8
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 688 in 130 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C4 [×2], C22, C22 [×8], S3 [×4], C6 [×10], C8 [×2], C2×C4, D4 [×6], C23 [×2], C32, Dic3 [×4], D6 [×8], C2×C6 [×10], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×4], C3×C6, C3×C6 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C22×S3 [×2], C22×C6 [×2], C2×D8, C3⋊Dic3 [×2], S3×C6 [×8], C62, C2×C3⋊D4 [×2], C322C8 [×2], D6⋊S3 [×4], D6⋊S3 [×2], C2×C3⋊Dic3, S3×C2×C6 [×2], C32⋊D8 [×4], C2×C322C8, C2×D6⋊S3 [×2], C2×C32⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, C2×D8, S3≀C2, C32⋊D8 [×2], C2×S3≀C2, C2×C32⋊D8

Character table of C2×C32⋊D8

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 8A 8B 8C 8D size 1 1 1 1 12 12 12 12 4 4 18 18 4 4 4 4 4 4 12 12 12 12 12 12 12 12 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ6 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ8 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 -2 -2 0 0 0 0 2 2 2 -2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 0 0 2 2 -2 -2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 0 2 2 0 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ12 2 -2 -2 2 0 0 0 0 2 2 0 0 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 0 0 2 2 0 0 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ14 2 -2 2 -2 0 0 0 0 2 2 0 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ15 4 4 4 4 -2 0 0 -2 -2 1 0 0 -2 1 1 -2 -2 1 0 0 0 1 1 0 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 4 4 2 0 0 2 -2 1 0 0 -2 1 1 -2 -2 1 0 0 0 -1 -1 0 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 4 4 4 0 -2 -2 0 1 -2 0 0 1 -2 -2 1 1 -2 1 1 1 0 0 1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ18 4 4 -4 -4 0 2 -2 0 1 -2 0 0 -1 2 -2 1 -1 2 -1 1 -1 0 0 1 0 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ19 4 4 -4 -4 2 0 0 -2 -2 1 0 0 2 -1 1 -2 2 -1 0 0 0 1 1 0 -1 -1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ20 4 4 -4 -4 -2 0 0 2 -2 1 0 0 2 -1 1 -2 2 -1 0 0 0 -1 -1 0 1 1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ21 4 4 -4 -4 0 -2 2 0 1 -2 0 0 -1 2 -2 1 -1 2 1 -1 1 0 0 -1 0 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ22 4 4 4 4 0 2 2 0 1 -2 0 0 1 -2 -2 1 1 -2 -1 -1 -1 0 0 -1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ23 4 -4 -4 4 0 0 0 0 -2 1 0 0 -2 -1 -1 2 2 1 0 0 0 √-3 -√-3 0 -√-3 √-3 0 0 0 0 complex lifted from C32⋊D8 ρ24 4 -4 4 -4 0 0 0 0 -2 1 0 0 2 1 -1 2 -2 -1 0 0 0 √-3 -√-3 0 √-3 -√-3 0 0 0 0 complex lifted from C32⋊D8 ρ25 4 -4 -4 4 0 0 0 0 1 -2 0 0 1 2 2 -1 -1 -2 √-3 √-3 -√-3 0 0 -√-3 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ26 4 -4 4 -4 0 0 0 0 1 -2 0 0 -1 -2 2 -1 1 2 √-3 -√-3 -√-3 0 0 √-3 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ27 4 -4 4 -4 0 0 0 0 1 -2 0 0 -1 -2 2 -1 1 2 -√-3 √-3 √-3 0 0 -√-3 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ28 4 -4 -4 4 0 0 0 0 1 -2 0 0 1 2 2 -1 -1 -2 -√-3 -√-3 √-3 0 0 √-3 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ29 4 -4 -4 4 0 0 0 0 -2 1 0 0 -2 -1 -1 2 2 1 0 0 0 -√-3 √-3 0 √-3 -√-3 0 0 0 0 complex lifted from C32⋊D8 ρ30 4 -4 4 -4 0 0 0 0 -2 1 0 0 2 1 -1 2 -2 -1 0 0 0 -√-3 √-3 0 -√-3 √-3 0 0 0 0 complex lifted from C32⋊D8

Smallest permutation representation of C2×C32⋊D8
On 48 points
Generators in S48
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 11 18)(3 20 13)(5 15 22)(7 24 9)(26 33 46)(28 48 35)(30 37 42)(32 44 39)
(2 19 12)(4 14 21)(6 23 16)(8 10 17)(25 40 45)(27 47 34)(29 36 41)(31 43 38)
(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)
(2 8)(3 7)(4 6)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 48)(34 40)(35 39)(36 38)

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

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

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

Matrix representation of C2×C32⋊D8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 33 50 0 0 0 0 22 40 0 0 0 0 0 0 0 0 30 13 0 0 0 0 60 43 0 0 0 1 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 60 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 30 13 0 0 0 0 60 43

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[33,22,0,0,0,0,50,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,30,60,0,0,0,0,13,43,0,0],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,30,60,0,0,0,0,13,43] >;

C2×C32⋊D8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes D_8
% in TeX

G:=Group("C2xC3^2:D8");
// GroupNames label

G:=SmallGroup(288,883);
// by ID

G=gap.SmallGroup(288,883);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,675,346,80,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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