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## G = C3×D4×Q8order 192 = 26·3

### Direct product of C3, D4 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×D4×Q8
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C6×Q8 — C3×C22⋊Q8 — C3×D4×Q8
 Lower central C1 — C22 — C3×D4×Q8
 Upper central C1 — C2×C6 — C3×D4×Q8

Generators and relations for C3×D4×Q8
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 378 in 280 conjugacy classes, 182 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C6×Q8, C6×Q8, D4×Q8, D4×C12, Q8×C12, C3×C22⋊Q8, C3×C4⋊Q8, Q8×C2×C6, C3×D4×Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C24, C3×D4, C3×Q8, C22×C6, C22×D4, C22×Q8, 2- 1+4, C6×D4, C6×Q8, C23×C6, D4×Q8, D4×C2×C6, Q8×C2×C6, C3×2- 1+4, C3×D4×Q8

Smallest permutation representation of C3×D4×Q8
On 96 points
Generators in S96
(1 27 83)(2 28 84)(3 25 81)(4 26 82)(5 53 15)(6 54 16)(7 55 13)(8 56 14)(9 37 71)(10 38 72)(11 39 69)(12 40 70)(17 50 22)(18 51 23)(19 52 24)(20 49 21)(29 85 89)(30 86 90)(31 87 91)(32 88 92)(33 61 93)(34 62 94)(35 63 95)(36 64 96)(41 75 79)(42 76 80)(43 73 77)(44 74 78)(45 67 58)(46 68 59)(47 65 60)(48 66 57)
(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 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 16)(14 15)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 38)(39 40)(41 44)(42 43)(45 48)(46 47)(49 52)(50 51)(53 56)(54 55)(57 58)(59 60)(61 64)(62 63)(65 68)(66 67)(69 70)(71 72)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)
(1 75 15 21)(2 76 16 22)(3 73 13 23)(4 74 14 24)(5 20 27 79)(6 17 28 80)(7 18 25 77)(8 19 26 78)(9 66 88 36)(10 67 85 33)(11 68 86 34)(12 65 87 35)(29 93 72 45)(30 94 69 46)(31 95 70 47)(32 96 71 48)(37 57 92 64)(38 58 89 61)(39 59 90 62)(40 60 91 63)(41 53 49 83)(42 54 50 84)(43 55 51 81)(44 56 52 82)
(1 35 15 65)(2 36 16 66)(3 33 13 67)(4 34 14 68)(5 60 27 63)(6 57 28 64)(7 58 25 61)(8 59 26 62)(9 76 88 22)(10 73 85 23)(11 74 86 24)(12 75 87 21)(17 37 80 92)(18 38 77 89)(19 39 78 90)(20 40 79 91)(29 51 72 43)(30 52 69 44)(31 49 70 41)(32 50 71 42)(45 81 93 55)(46 82 94 56)(47 83 95 53)(48 84 96 54)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4H 4I ··· 4Q 6A ··· 6F 6G ··· 6N 12A ··· 12P 12Q ··· 12AH order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 2 2 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + - + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 Q8 D4 C3×Q8 C3×D4 2- 1+4 C3×2- 1+4 kernel C3×D4×Q8 D4×C12 Q8×C12 C3×C22⋊Q8 C3×C4⋊Q8 Q8×C2×C6 D4×Q8 C4×D4 C4×Q8 C22⋊Q8 C4⋊Q8 C22×Q8 C3×D4 C3×Q8 D4 Q8 C6 C2 # reps 1 3 1 6 3 2 2 6 2 12 6 4 4 4 8 8 1 2

Matrix representation of C3×D4×Q8 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 12 0
,
 8 0 0 0 5 5 0 0 0 0 12 0 0 0 0 12
,
 12 11 0 0 1 1 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,12,0],[8,5,0,0,0,5,0,0,0,0,12,0,0,0,0,12],[12,1,0,0,11,1,0,0,0,0,12,0,0,0,0,12] >;

C3×D4×Q8 in GAP, Magma, Sage, TeX

C_3\times D_4\times Q_8
% in TeX

G:=Group("C3xD4xQ8");
// GroupNames label

G:=SmallGroup(192,1438);
// by ID

G=gap.SmallGroup(192,1438);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,344,2102,794,192]);
// Polycyclic

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

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