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## G = C3×Q8⋊C4order 96 = 25·3

### Direct product of C3 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×Q8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×Q8⋊C4
 Lower central C1 — C2 — C4 — C3×Q8⋊C4
 Upper central C1 — C2×C6 — C2×C12 — C3×Q8⋊C4

Generators and relations for C3×Q8⋊C4
G = < a,b,c,d | a3=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Smallest permutation representation of C3×Q8⋊C4
Regular action on 96 points
Generators in S96
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 91 83)(6 92 84)(7 89 81)(8 90 82)(9 28 17)(10 25 18)(11 26 19)(12 27 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 74 65)(58 75 66)(59 76 67)(60 73 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(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 34 3 36)(2 33 4 35)(5 71 7 69)(6 70 8 72)(9 32 11 30)(10 31 12 29)(13 41 15 43)(14 44 16 42)(17 40 19 38)(18 39 20 37)(21 49 23 51)(22 52 24 50)(25 47 27 45)(26 46 28 48)(53 83 55 81)(54 82 56 84)(57 80 59 78)(58 79 60 77)(61 91 63 89)(62 90 64 92)(65 88 67 86)(66 87 68 85)(73 93 75 95)(74 96 76 94)
(1 60 11 56)(2 59 12 55)(3 58 9 54)(4 57 10 53)(5 52 96 48)(6 51 93 47)(7 50 94 46)(8 49 95 45)(13 67 20 63)(14 66 17 62)(15 65 18 61)(16 68 19 64)(21 76 27 71)(22 75 28 70)(23 74 25 69)(24 73 26 72)(29 82 33 79)(30 81 34 78)(31 84 35 77)(32 83 36 80)(37 90 41 87)(38 89 42 86)(39 92 43 85)(40 91 44 88)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L 24A ··· 24H order 1 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 2 2 2 2 4 ··· 4 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 SD16 Q16 C3×D4 C3×D4 C3×SD16 C3×Q16 kernel C3×Q8⋊C4 C3×C4⋊C4 C2×C24 C6×Q8 Q8⋊C4 C3×Q8 C4⋊C4 C2×C8 C2×Q8 Q8 C12 C2×C6 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 4 4

Matrix representation of C3×Q8⋊C4 in GL3(𝔽73) generated by

 64 0 0 0 8 0 0 0 8
,
 1 0 0 0 0 1 0 72 0
,
 1 0 0 0 65 9 0 9 8
,
 46 0 0 0 6 29 0 29 67
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[1,0,0,0,0,72,0,1,0],[1,0,0,0,65,9,0,9,8],[46,0,0,0,6,29,0,29,67] >;

C3×Q8⋊C4 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes C_4
% in TeX

G:=Group("C3xQ8:C4");
// GroupNames label

G:=SmallGroup(96,53);
// by ID

G=gap.SmallGroup(96,53);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,295,1443,729,117]);
// Polycyclic

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

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