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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 — C22 — C2×C4 — C2×C12 — C6×D4 — C6×D8 — C3×D4⋊D4
 Lower central C1 — C2 — C2×C4 — C3×D4⋊D4
 Upper central C1 — C2×C6 — C22×C12 — C3×D4⋊D4

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

Subgroups: 322 in 162 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×D8, C3×SD16, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, D4⋊D4, C3×C22⋊C8, C3×D4⋊C4, C3×Q8⋊C4, C3×C4⋊D4, C6×D8, C6×SD16, C6×C4○D4, C3×D4⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C4○D8, C8⋊C22, C6×D4, D4⋊D4, C3×C22≀C2, C3×C4○D8, C3×C8⋊C22, C3×D4⋊D4

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

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G ··· 6L 6M 6N 8A 8B 8C 8D 12A ··· 12H 12I 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 2 2 2 2 3 3 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 4 8 1 1 2 2 2 2 4 4 8 1 ··· 1 4 ··· 4 8 8 4 4 4 4 2 ··· 2 4 4 4 4 8 8 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 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 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 D4 D4 C3×D4 C3×D4 C3×D4 C3×D4 C4○D8 C3×C4○D8 C8⋊C22 C3×C8⋊C22 kernel C3×D4⋊D4 C3×C22⋊C8 C3×D4⋊C4 C3×Q8⋊C4 C3×C4⋊D4 C6×D8 C6×SD16 C6×C4○D4 D4⋊D4 C22⋊C8 D4⋊C4 Q8⋊C4 C4⋊D4 C2×D8 C2×SD16 C2×C4○D4 C2×C12 C3×D4 C3×Q8 C22×C6 C2×C4 D4 Q8 C23 C6 C2 C6 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 2 2 1 2 4 4 2 4 8 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
,
 1 71 0 0 1 72 0 0 0 0 1 0 0 0 0 1
,
 0 32 0 0 16 0 0 0 0 0 72 0 0 0 0 72
,
 27 19 0 0 0 46 0 0 0 0 0 72 0 0 1 0
,
 1 0 0 0 1 72 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[1,1,0,0,71,72,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,32,0,0,0,0,0,72,0,0,0,0,72],[27,0,0,0,19,46,0,0,0,0,0,1,0,0,72,0],[1,1,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;

C3×D4⋊D4 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes D_4
% in TeX

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

G:=SmallGroup(192,882);
// by ID

G=gap.SmallGroup(192,882);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,520,4204,2111,172]);
// Polycyclic

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

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