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G = C3×C4⋊D8order 192 = 26·3

Direct product of C3 and C4⋊D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C4⋊D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×D4 — C3×C4⋊1D4 — C3×C4⋊D8
 Lower central C1 — C2 — C2×C4 — C3×C4⋊D8
 Upper central C1 — C2×C6 — C4×C12 — C3×C4⋊D8

Generators and relations for C3×C4⋊D8
G = < a,b,c,d | a3=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 314 in 140 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C12, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C4⋊C8, C4×D4, C41D4, C2×D8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×D8, C22×C12, C6×D4, C6×D4, C6×D4, C4⋊D8, C3×D4⋊C4, C3×C4⋊C8, D4×C12, C3×C41D4, C6×D8, C3×C4⋊D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C2×D8, C8⋊C22, C3×D8, C6×D4, C3×C4○D4, C4⋊D8, C3×C4⋊D4, C6×D8, C3×C8⋊C22, C3×C4⋊D8

Smallest permutation representation of C3×C4⋊D8
On 96 points
Generators in S96
(1 49 28)(2 50 29)(3 51 30)(4 52 31)(5 53 32)(6 54 25)(7 55 26)(8 56 27)(9 67 83)(10 68 84)(11 69 85)(12 70 86)(13 71 87)(14 72 88)(15 65 81)(16 66 82)(17 76 91)(18 77 92)(19 78 93)(20 79 94)(21 80 95)(22 73 96)(23 74 89)(24 75 90)(33 46 63)(34 47 64)(35 48 57)(36 41 58)(37 42 59)(38 43 60)(39 44 61)(40 45 62)
(1 66 19 62)(2 63 20 67)(3 68 21 64)(4 57 22 69)(5 70 23 58)(6 59 24 71)(7 72 17 60)(8 61 18 65)(9 29 46 94)(10 95 47 30)(11 31 48 96)(12 89 41 32)(13 25 42 90)(14 91 43 26)(15 27 44 92)(16 93 45 28)(33 79 83 50)(34 51 84 80)(35 73 85 52)(36 53 86 74)(37 75 87 54)(38 55 88 76)(39 77 81 56)(40 49 82 78)
(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 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 64)(8 63)(9 92)(10 91)(11 90)(12 89)(13 96)(14 95)(15 94)(16 93)(17 68)(18 67)(19 66)(20 65)(21 72)(22 71)(23 70)(24 69)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(80 88)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 8A 8B 8C 8D 12A ··· 12H 12I ··· 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 6 6 6 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 8 8 1 1 2 2 2 2 4 4 4 1 ··· 1 4 4 4 4 8 8 8 8 4 4 4 4 2 ··· 2 4 ··· 4 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 D8 C4○D4 C3×D4 C3×D4 C3×D8 C3×C4○D4 C8⋊C22 C3×C8⋊C22 kernel C3×C4⋊D8 C3×D4⋊C4 C3×C4⋊C8 D4×C12 C3×C4⋊1D4 C6×D8 C4⋊D8 D4⋊C4 C4⋊C8 C4×D4 C4⋊1D4 C2×D8 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×C4⋊D8 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 64 0 0 0 0 64
,
 0 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 57 16 0 0 57 57
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,57,57,0,0,16,57],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,72] >;

C3×C4⋊D8 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes D_8
% in TeX

G:=Group("C3xC4:D8");
// GroupNames label

G:=SmallGroup(192,892);
// by ID

G=gap.SmallGroup(192,892);
# by ID

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

G:=Group<a,b,c,d|a^3=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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