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## G = C3×C8⋊4D4order 192 = 26·3

### Direct product of C3 and C8⋊4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8⋊4D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×D4 — C6×D8 — C3×C8⋊4D4
 Lower central C1 — C2 — C2×C4 — C3×C8⋊4D4
 Upper central C1 — C2×C6 — C4×C12 — C3×C8⋊4D4

Generators and relations for C3×C84D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 386 in 162 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, C2×C6, C2×C6, C42, C2×C8, D8, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C4×C8, C41D4, C2×D8, C4×C12, C2×C24, C3×D8, C6×D4, C6×D4, C84D4, C4×C24, C3×C41D4, C6×D8, C3×C84D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C41D4, C2×D8, C3×D8, C6×D4, C84D4, C3×C41D4, C6×D8, C3×C84D4

Smallest permutation representation of C3×C84D4
On 96 points
Generators in S96
(1 83 33)(2 84 34)(3 85 35)(4 86 36)(5 87 37)(6 88 38)(7 81 39)(8 82 40)(9 17 59)(10 18 60)(11 19 61)(12 20 62)(13 21 63)(14 22 64)(15 23 57)(16 24 58)(25 76 66)(26 77 67)(27 78 68)(28 79 69)(29 80 70)(30 73 71)(31 74 72)(32 75 65)(41 50 91)(42 51 92)(43 52 93)(44 53 94)(45 54 95)(46 55 96)(47 56 89)(48 49 90)
(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 65 13 41)(2 66 14 42)(3 67 15 43)(4 68 16 44)(5 69 9 45)(6 70 10 46)(7 71 11 47)(8 72 12 48)(17 54 87 28)(18 55 88 29)(19 56 81 30)(20 49 82 31)(21 50 83 32)(22 51 84 25)(23 52 85 26)(24 53 86 27)(33 75 63 91)(34 76 64 92)(35 77 57 93)(36 78 58 94)(37 79 59 95)(38 80 60 96)(39 73 61 89)(40 74 62 90)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 40)(34 39)(35 38)(36 37)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(57 60)(58 59)(61 64)(62 63)(73 92)(74 91)(75 90)(76 89)(77 96)(78 95)(79 94)(80 93)(81 84)(82 83)(85 88)(86 87)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4F 6A ··· 6F 6G ··· 6N 8A ··· 8H 12A ··· 12L 24A ··· 24P order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 8 8 8 8 1 1 2 ··· 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2 2 ··· 2

66 irreducible representations

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

Matrix representation of C3×C84D4 in GL4(𝔽73) generated by

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

C3×C84D4 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_4D_4
% in TeX

G:=Group("C3xC8:4D4");
// GroupNames label

G:=SmallGroup(192,926);
// by ID

G=gap.SmallGroup(192,926);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,268,4204,172]);
// Polycyclic

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

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