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## G = C3×Dic3⋊Q8order 288 = 25·32

### Direct product of C3 and Dic3⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×Dic3⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C12 — C3×Dic3⋊Q8
 Lower central C3 — C2×C6 — C3×Dic3⋊Q8
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C3×Dic3⋊Q8
G = < a,b,c,d,e | a3=b6=d4=1, c2=b3, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 298 in 155 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×8], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C32, Dic3 [×4], Dic3 [×2], C12 [×4], C12 [×16], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8, C2×Q8, C3×C6, C3×C6 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C3×Q8 [×10], C4⋊Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×4], C4×C12, C3×C4⋊C4 [×4], C2×Dic6, C6×Q8 [×2], C6×Q8 [×2], C3×Dic6 [×2], C6×Dic3 [×4], C6×C12, C6×C12 [×2], Q8×C32 [×2], Dic3⋊Q8, C3×C4⋊Q8, Dic3×C12, C3×Dic3⋊C4 [×4], C6×Dic6, Q8×C3×C6, C3×Dic3⋊Q8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], Q8 [×4], C23, D6 [×3], C2×C6 [×7], C2×D4, C2×Q8 [×2], C3×S3, C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×4], C22×S3, C22×C6, C4⋊Q8, S3×C6 [×3], S3×Q8 [×2], C2×C3⋊D4, C6×D4, C6×Q8 [×2], C3×C3⋊D4 [×2], S3×C2×C6, Dic3⋊Q8, C3×C4⋊Q8, C3×S3×Q8 [×2], C6×C3⋊D4, C3×Dic3⋊Q8

Smallest permutation representation of C3×Dic3⋊Q8
On 96 points
Generators in S96
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 83 4 80)(2 82 5 79)(3 81 6 84)(7 19 10 22)(8 24 11 21)(9 23 12 20)(13 85 16 88)(14 90 17 87)(15 89 18 86)(25 96 28 93)(26 95 29 92)(27 94 30 91)(31 59 34 56)(32 58 35 55)(33 57 36 60)(37 62 40 65)(38 61 41 64)(39 66 42 63)(43 71 46 68)(44 70 47 67)(45 69 48 72)(49 75 52 78)(50 74 53 77)(51 73 54 76)
(1 19 13 26)(2 20 14 27)(3 21 15 28)(4 22 16 29)(5 23 17 30)(6 24 18 25)(7 85 92 83)(8 86 93 84)(9 87 94 79)(10 88 95 80)(11 89 96 81)(12 90 91 82)(31 43 38 51)(32 44 39 52)(33 45 40 53)(34 46 41 54)(35 47 42 49)(36 48 37 50)(55 70 63 78)(56 71 64 73)(57 72 65 74)(58 67 66 75)(59 68 61 76)(60 69 62 77)
(1 31 13 38)(2 32 14 39)(3 33 15 40)(4 34 16 41)(5 35 17 42)(6 36 18 37)(7 76 92 68)(8 77 93 69)(9 78 94 70)(10 73 95 71)(11 74 96 72)(12 75 91 67)(19 51 26 43)(20 52 27 44)(21 53 28 45)(22 54 29 46)(23 49 30 47)(24 50 25 48)(55 87 63 79)(56 88 64 80)(57 89 65 81)(58 90 66 82)(59 85 61 83)(60 86 62 84)

G:=sub<Sym(96)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,83,4,80)(2,82,5,79)(3,81,6,84)(7,19,10,22)(8,24,11,21)(9,23,12,20)(13,85,16,88)(14,90,17,87)(15,89,18,86)(25,96,28,93)(26,95,29,92)(27,94,30,91)(31,59,34,56)(32,58,35,55)(33,57,36,60)(37,62,40,65)(38,61,41,64)(39,66,42,63)(43,71,46,68)(44,70,47,67)(45,69,48,72)(49,75,52,78)(50,74,53,77)(51,73,54,76), (1,19,13,26)(2,20,14,27)(3,21,15,28)(4,22,16,29)(5,23,17,30)(6,24,18,25)(7,85,92,83)(8,86,93,84)(9,87,94,79)(10,88,95,80)(11,89,96,81)(12,90,91,82)(31,43,38,51)(32,44,39,52)(33,45,40,53)(34,46,41,54)(35,47,42,49)(36,48,37,50)(55,70,63,78)(56,71,64,73)(57,72,65,74)(58,67,66,75)(59,68,61,76)(60,69,62,77), (1,31,13,38)(2,32,14,39)(3,33,15,40)(4,34,16,41)(5,35,17,42)(6,36,18,37)(7,76,92,68)(8,77,93,69)(9,78,94,70)(10,73,95,71)(11,74,96,72)(12,75,91,67)(19,51,26,43)(20,52,27,44)(21,53,28,45)(22,54,29,46)(23,49,30,47)(24,50,25,48)(55,87,63,79)(56,88,64,80)(57,89,65,81)(58,90,66,82)(59,85,61,83)(60,86,62,84)>;

G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,83,4,80)(2,82,5,79)(3,81,6,84)(7,19,10,22)(8,24,11,21)(9,23,12,20)(13,85,16,88)(14,90,17,87)(15,89,18,86)(25,96,28,93)(26,95,29,92)(27,94,30,91)(31,59,34,56)(32,58,35,55)(33,57,36,60)(37,62,40,65)(38,61,41,64)(39,66,42,63)(43,71,46,68)(44,70,47,67)(45,69,48,72)(49,75,52,78)(50,74,53,77)(51,73,54,76), (1,19,13,26)(2,20,14,27)(3,21,15,28)(4,22,16,29)(5,23,17,30)(6,24,18,25)(7,85,92,83)(8,86,93,84)(9,87,94,79)(10,88,95,80)(11,89,96,81)(12,90,91,82)(31,43,38,51)(32,44,39,52)(33,45,40,53)(34,46,41,54)(35,47,42,49)(36,48,37,50)(55,70,63,78)(56,71,64,73)(57,72,65,74)(58,67,66,75)(59,68,61,76)(60,69,62,77), (1,31,13,38)(2,32,14,39)(3,33,15,40)(4,34,16,41)(5,35,17,42)(6,36,18,37)(7,76,92,68)(8,77,93,69)(9,78,94,70)(10,73,95,71)(11,74,96,72)(12,75,91,67)(19,51,26,43)(20,52,27,44)(21,53,28,45)(22,54,29,46)(23,49,30,47)(24,50,25,48)(55,87,63,79)(56,88,64,80)(57,89,65,81)(58,90,66,82)(59,85,61,83)(60,86,62,84) );

G=PermutationGroup([(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46),(49,53,51),(50,54,52),(55,57,59),(56,58,60),(61,63,65),(62,64,66),(67,69,71),(68,70,72),(73,75,77),(74,76,78),(79,81,83),(80,82,84),(85,87,89),(86,88,90),(91,93,95),(92,94,96)], [(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,83,4,80),(2,82,5,79),(3,81,6,84),(7,19,10,22),(8,24,11,21),(9,23,12,20),(13,85,16,88),(14,90,17,87),(15,89,18,86),(25,96,28,93),(26,95,29,92),(27,94,30,91),(31,59,34,56),(32,58,35,55),(33,57,36,60),(37,62,40,65),(38,61,41,64),(39,66,42,63),(43,71,46,68),(44,70,47,67),(45,69,48,72),(49,75,52,78),(50,74,53,77),(51,73,54,76)], [(1,19,13,26),(2,20,14,27),(3,21,15,28),(4,22,16,29),(5,23,17,30),(6,24,18,25),(7,85,92,83),(8,86,93,84),(9,87,94,79),(10,88,95,80),(11,89,96,81),(12,90,91,82),(31,43,38,51),(32,44,39,52),(33,45,40,53),(34,46,41,54),(35,47,42,49),(36,48,37,50),(55,70,63,78),(56,71,64,73),(57,72,65,74),(58,67,66,75),(59,68,61,76),(60,69,62,77)], [(1,31,13,38),(2,32,14,39),(3,33,15,40),(4,34,16,41),(5,35,17,42),(6,36,18,37),(7,76,92,68),(8,77,93,69),(9,78,94,70),(10,73,95,71),(11,74,96,72),(12,75,91,67),(19,51,26,43),(20,52,27,44),(21,53,28,45),(22,54,29,46),(23,49,30,47),(24,50,25,48),(55,87,63,79),(56,88,64,80),(57,89,65,81),(58,90,66,82),(59,85,61,83),(60,86,62,84)])

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6O 12A 12B 12C 12D 12E ··· 12Z 12AA ··· 12AH 12AI 12AJ 12AK 12AL order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 1 1 2 2 2 2 2 4 4 6 6 6 6 12 12 1 ··· 1 2 ··· 2 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 Q8 D4 D6 C3×S3 C3×Q8 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 S3×Q8 C3×S3×Q8 kernel C3×Dic3⋊Q8 Dic3×C12 C3×Dic3⋊C4 C6×Dic6 Q8×C3×C6 Dic3⋊Q8 C4×Dic3 Dic3⋊C4 C2×Dic6 C6×Q8 C6×Q8 C3×Dic3 C3×C12 C2×C12 C2×Q8 Dic3 C12 C12 C2×C4 C4 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 4 2 3 2 8 4 4 6 8 2 4

Matrix representation of C3×Dic3⋊Q8 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 10 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 0 12 0 0 1 0 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 12 5 0 0 10 1
,
 1 0 0 0 0 1 0 0 0 0 5 0 0 0 2 8
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[10,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,10,0,0,5,1],[1,0,0,0,0,1,0,0,0,0,5,2,0,0,0,8] >;

C3×Dic3⋊Q8 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\rtimes Q_8
% in TeX

G:=Group("C3xDic3:Q8");
// GroupNames label

G:=SmallGroup(288,715);
// by ID

G=gap.SmallGroup(288,715);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,1094,303,142,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=d^4=1,c^2=b^3,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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