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## G = C3×D4⋊3Q8order 192 = 26·3

### Direct product of C3 and D4⋊3Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×D4⋊3Q8
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C4⋊C4 — C3×C22⋊Q8 — C3×D4⋊3Q8
 Lower central C1 — C22 — C3×D4⋊3Q8
 Upper central C1 — C2×C6 — C3×D4⋊3Q8

Generators and relations for C3×D43Q8
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, ece-1=b2c, ede-1=d-1 >

Subgroups: 314 in 228 conjugacy classes, 166 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C6×Q8, D43Q8, C6×C4⋊C4, D4×C12, D4×C12, Q8×C12, C3×C22⋊Q8, C3×C42.C2, C3×C4⋊Q8, C3×D43Q8
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C4○D4, C24, C3×Q8, C22×C6, C22×Q8, C2×C4○D4, 2+ 1+4, C6×Q8, C3×C4○D4, C23×C6, D43Q8, Q8×C2×C6, C6×C4○D4, C3×2+ 1+4, C3×D43Q8

Smallest permutation representation of C3×D43Q8
On 96 points
Generators in S96
(1 27 83)(2 28 84)(3 25 81)(4 26 82)(5 51 23)(6 52 24)(7 49 21)(8 50 22)(9 37 71)(10 38 72)(11 39 69)(12 40 70)(13 57 19)(14 58 20)(15 59 17)(16 60 18)(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 56)(46 68 53)(47 65 54)(48 66 55)
(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 20)(18 19)(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 54)(55 56)(57 60)(58 59)(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 13 23)(2 76 14 24)(3 73 15 21)(4 74 16 22)(5 27 79 57)(6 28 80 58)(7 25 77 59)(8 26 78 60)(9 68 88 36)(10 65 85 33)(11 66 86 34)(12 67 87 35)(17 49 81 43)(18 50 82 44)(19 51 83 41)(20 52 84 42)(29 93 72 47)(30 94 69 48)(31 95 70 45)(32 96 71 46)(37 53 92 64)(38 54 89 61)(39 55 90 62)(40 56 91 63)
(1 34 13 66)(2 35 14 67)(3 36 15 68)(4 33 16 65)(5 39 79 90)(6 40 80 91)(7 37 77 92)(8 38 78 89)(9 73 88 21)(10 74 85 22)(11 75 86 23)(12 76 87 24)(17 46 81 96)(18 47 82 93)(19 48 83 94)(20 45 84 95)(25 64 59 53)(26 61 60 54)(27 62 57 55)(28 63 58 56)(29 50 72 44)(30 51 69 41)(31 52 70 42)(32 49 71 43)

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

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

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

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 1 1 2 2 2 2 4 4 type + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 Q8 C4○D4 C3×Q8 C3×C4○D4 2+ 1+4 C3×2+ 1+4 kernel C3×D4⋊3Q8 C6×C4⋊C4 D4×C12 Q8×C12 C3×C22⋊Q8 C3×C42.C2 C3×C4⋊Q8 D4⋊3Q8 C2×C4⋊C4 C4×D4 C4×Q8 C22⋊Q8 C42.C2 C4⋊Q8 C3×D4 C12 D4 C4 C6 C2 # reps 1 2 3 1 6 2 1 2 4 6 2 12 4 2 4 4 8 8 1 2

Matrix representation of C3×D43Q8 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 12 11 0 0 1 1
,
 1 0 0 0 0 1 0 0 0 0 12 11 0 0 0 1
,
 7 2 0 0 1 6 0 0 0 0 1 0 0 0 0 1
,
 4 6 0 0 8 9 0 0 0 0 8 3 0 0 5 5
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,11,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,11,1],[7,1,0,0,2,6,0,0,0,0,1,0,0,0,0,1],[4,8,0,0,6,9,0,0,0,0,8,5,0,0,3,5] >;

C3×D43Q8 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_3Q_8
% in TeX

G:=Group("C3xD4:3Q8");
// GroupNames label

G:=SmallGroup(192,1443);
// by ID

G=gap.SmallGroup(192,1443);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,1016,2102,794,192]);
// 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,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations

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