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## G = C3×D4.D4order 192 = 26·3

### Direct product of C3 and D4.D4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×D4.D4
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 218 in 120 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C22×C12, C6×D4, C6×Q8, D4.D4, C3×Q8⋊C4, C3×C4⋊C8, D4×C12, C3×C4⋊Q8, C6×SD16, C3×D4.D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C2×SD16, C8.C22, C3×SD16, C6×D4, C3×C4○D4, D4.D4, C3×C4⋊D4, C6×SD16, C3×C8.C22, C3×D4.D4

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

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A ··· 12H 12I ··· 12N 12O 12P 12Q 12R 24A ··· 24H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 8 8 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 4 4 1 1 2 2 2 2 4 4 4 8 8 1 ··· 1 4 4 4 4 4 4 4 4 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

57 irreducible representations

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

Matrix representation of C3×D4.D4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 72 0 0 0 0 72 0 0 0 0 1 71 0 0 1 72
,
 72 31 0 0 0 1 0 0 0 0 1 71 0 0 0 72
,
 27 39 0 0 0 46 0 0 0 0 1 0 0 0 0 1
,
 5 57 0 0 38 68 0 0 0 0 0 12 0 0 6 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,1,1,0,0,71,72],[72,0,0,0,31,1,0,0,0,0,1,0,0,0,71,72],[27,0,0,0,39,46,0,0,0,0,1,0,0,0,0,1],[5,38,0,0,57,68,0,0,0,0,0,6,0,0,12,0] >;

C3×D4.D4 in GAP, Magma, Sage, TeX

C_3\times D_4.D_4
% in TeX

G:=Group("C3xD4.D4");
// GroupNames label

G:=SmallGroup(192,894);
// by ID

G=gap.SmallGroup(192,894);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,176,1094,6053,1531,124]);
// Polycyclic

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

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