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## G = C3×D4⋊C8order 192 = 26·3

### Direct product of C3 and D4⋊C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×D4⋊C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C12 — C3×C4⋊C8 — C3×D4⋊C8
 Lower central C1 — C2 — C4 — C3×D4⋊C8
 Upper central C1 — C2×C12 — C4×C12 — C3×D4⋊C8

Generators and relations for C3×D4⋊C8
G = < a,b,c,d | a3=b4=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 154 in 82 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C4×C8, C4⋊C8, C4×D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C6×D4, D4⋊C8, C4×C24, C3×C4⋊C8, D4×C12, C3×D4⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C8, M4(2), D8, SD16, C24, C2×C12, C3×D4, C22⋊C8, D4⋊C4, C4≀C2, C3×C22⋊C4, C2×C24, C3×M4(2), C3×D8, C3×SD16, D4⋊C8, C3×C22⋊C8, C3×D4⋊C4, C3×C4≀C2, C3×D4⋊C8

Smallest permutation representation of C3×D4⋊C8
On 96 points
Generators in S96
(1 10 65)(2 11 66)(3 12 67)(4 13 68)(5 14 69)(6 15 70)(7 16 71)(8 9 72)(17 63 25)(18 64 26)(19 57 27)(20 58 28)(21 59 29)(22 60 30)(23 61 31)(24 62 32)(33 55 41)(34 56 42)(35 49 43)(36 50 44)(37 51 45)(38 52 46)(39 53 47)(40 54 48)(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 39 79 17)(2 18 80 40)(3 33 73 19)(4 20 74 34)(5 35 75 21)(6 22 76 36)(7 37 77 23)(8 24 78 38)(9 62 94 52)(10 53 95 63)(11 64 96 54)(12 55 89 57)(13 58 90 56)(14 49 91 59)(15 60 92 50)(16 51 93 61)(25 65 47 87)(26 88 48 66)(27 67 41 81)(28 82 42 68)(29 69 43 83)(30 84 44 70)(31 71 45 85)(32 86 46 72)
(1 17)(2 80)(3 19)(4 74)(5 21)(6 76)(7 23)(8 78)(9 94)(10 63)(11 96)(12 57)(13 90)(14 59)(15 92)(16 61)(25 65)(27 67)(29 69)(31 71)(33 73)(35 75)(37 77)(39 79)(41 81)(43 83)(45 85)(47 87)(49 91)(51 93)(53 95)(55 89)(66 88)(68 82)(70 84)(72 86)
(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)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G 6H 6I 6J 8A ··· 8H 8I 8J 8K 8L 12A ··· 12H 12I ··· 12P 12Q 12R 12S 12T 24A ··· 24P 24Q ··· 24X order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 8 ··· 8 8 8 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 ··· 24 24 ··· 24 size 1 1 1 1 4 4 1 1 1 1 1 1 2 2 2 2 4 4 1 ··· 1 4 4 4 4 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 D4 M4(2) D8 SD16 C3×D4 C4≀C2 C3×M4(2) C3×D8 C3×SD16 C3×C4≀C2 kernel C3×D4⋊C8 C4×C24 C3×C4⋊C8 D4×C12 D4⋊C8 C3×C4⋊C4 C6×D4 C4×C8 C4⋊C8 C4×D4 C3×D4 C4⋊C4 C2×D4 D4 C2×C12 C12 C12 C12 C2×C4 C6 C4 C4 C4 C2 # reps 1 1 1 1 2 2 2 2 2 2 8 4 4 16 2 2 2 2 4 4 4 4 4 8

Matrix representation of C3×D4⋊C8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 72 0 0 0 0 72 0 0 0 0 1 72 0 0 2 72
,
 72 0 0 0 11 1 0 0 0 0 72 0 0 0 71 1
,
 63 38 0 0 0 10 0 0 0 0 61 6 0 0 61 12
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[72,0,0,0,0,72,0,0,0,0,1,2,0,0,72,72],[72,11,0,0,0,1,0,0,0,0,72,71,0,0,0,1],[63,0,0,0,38,10,0,0,0,0,61,61,0,0,6,12] >;

C3×D4⋊C8 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes C_8
% in TeX

G:=Group("C3xD4:C8");
// GroupNames label

G:=SmallGroup(192,131);
// by ID

G=gap.SmallGroup(192,131);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,850,136,172]);
// Polycyclic

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

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