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## G = C2×C3⋊S3.Q8order 288 = 25·32

### Direct product of C2 and C3⋊S3.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C2×C3⋊S3.Q8
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C6.D6 — C3⋊S3.Q8 — C2×C3⋊S3.Q8
 Lower central C32 — C3⋊S3 — C2×C3⋊S3.Q8
 Upper central C1 — C22

Generators and relations for C2×C3⋊S3.Q8
G = < a,b,c,d,e,f | a2=b3=c3=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd=ece-1=b-1, ebe-1=dcd=fcf-1=c-1, bf=fb, de=ed, df=fd, fef-1=de-1 >

Subgroups: 688 in 146 conjugacy classes, 43 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C4 [×8], C22, C22 [×6], S3 [×8], C6 [×6], C2×C4 [×14], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C4⋊C4 [×4], C22×C4 [×3], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2×C4⋊C4, C3×Dic3 [×4], C32⋊C4 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62, S3×C2×C4 [×2], C6.D6 [×4], C6.D6 [×2], C6×Dic3 [×2], C2×C32⋊C4 [×6], C22×C3⋊S3, C3⋊S3.Q8 [×4], C2×C6.D6 [×2], C22×C32⋊C4, C2×C3⋊S3.Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, S3≀C2, C3⋊S3.Q8 [×2], C2×S3≀C2, C2×C3⋊S3.Q8

Smallest permutation representation of C2×C3⋊S3.Q8
On 48 points
Generators in S48
(1 17)(2 18)(3 19)(4 20)(5 28)(6 25)(7 26)(8 27)(9 21)(10 22)(11 23)(12 24)(13 34)(14 35)(15 36)(16 33)(29 43)(30 44)(31 41)(32 42)(37 48)(38 45)(39 46)(40 47)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 43 47)(14 48 44)(15 41 45)(16 46 42)(17 28 21)(18 25 22)(19 26 23)(20 27 24)(29 40 34)(30 35 37)(31 38 36)(32 33 39)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 43 47)(14 48 44)(15 41 45)(16 46 42)(17 21 28)(18 22 25)(19 23 26)(20 24 27)(29 40 34)(30 35 37)(31 38 36)(32 33 39)
(5 9)(6 10)(7 11)(8 12)(13 47)(14 48)(15 45)(16 46)(21 28)(22 25)(23 26)(24 27)(33 39)(34 40)(35 37)(36 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)
(1 41 3 43)(2 44 4 42)(5 45 7 47)(6 14 8 16)(9 15 11 13)(10 48 12 46)(17 31 19 29)(18 30 20 32)(21 36 23 34)(22 37 24 39)(25 35 27 33)(26 40 28 38)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4H 4I 4J 4K 4L 6A ··· 6F 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 9 9 9 9 4 4 6 ··· 6 18 18 18 18 4 ··· 4 12 ··· 12

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 4 4 type + + + + + - + + + image C1 C2 C2 C2 C4 D4 Q8 D4 S3≀C2 C3⋊S3.Q8 C2×S3≀C2 kernel C2×C3⋊S3.Q8 C3⋊S3.Q8 C2×C6.D6 C22×C32⋊C4 C2×C32⋊C4 C2×C3⋊S3 C2×C3⋊S3 C62 C22 C2 C2 # reps 1 4 2 1 8 1 2 1 4 8 4

Matrix representation of C2×C3⋊S3.Q8 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 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 1 0
,
 0 8 0 0 0 0 5 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 6 2 0 0 0 0 2 7 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[6,2,0,0,0,0,2,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0] >;

C2×C3⋊S3.Q8 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes S_3.Q_8
% in TeX

G:=Group("C2xC3:S3.Q8");
// GroupNames label

G:=SmallGroup(288,882);
// by ID

G=gap.SmallGroup(288,882);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d=e*c*e^-1=b^-1,e*b*e^-1=d*c*d=f*c*f^-1=c^-1,b*f=f*b,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
// generators/relations

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