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## G = C3×C8.12D4order 192 = 26·3

### Direct product of C3 and C8.12D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8.12D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×D4 — C6×SD16 — C3×C8.12D4
 Lower central C1 — C2 — C2×C4 — C3×C8.12D4
 Upper central C1 — C2×C6 — C4×C12 — C3×C8.12D4

Generators and relations for C3×C8.12D4
G = < a,b,c,d | a3=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >

Subgroups: 258 in 130 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C4×C12, C3×C22⋊C4, C2×C24, C3×D8, C3×SD16, C3×Q16, C6×D4, C6×Q8, C8.12D4, C4×C24, C3×C4.4D4, C6×D8, C6×SD16, C6×Q16, C3×C8.12D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C41D4, C4○D8, C6×D4, C8.12D4, C3×C41D4, C3×C4○D8, C3×C8.12D4

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C4○D8 C3×C4○D8 kernel C3×C8.12D4 C4×C24 C3×C4.4D4 C6×D8 C6×SD16 C6×Q16 C8.12D4 C4×C8 C4.4D4 C2×D8 C2×SD16 C2×Q16 C24 C2×C12 C8 C2×C4 C6 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 4 2 8 4 8 16

Matrix representation of C3×C8.12D4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 27 71 0 0 0 46 0 0 0 0 6 67 0 0 6 6
,
 46 2 0 0 0 27 0 0 0 0 0 27 0 0 46 0
,
 27 71 0 0 72 46 0 0 0 0 67 6 0 0 6 6
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[27,0,0,0,71,46,0,0,0,0,6,6,0,0,67,6],[46,0,0,0,2,27,0,0,0,0,0,46,0,0,27,0],[27,72,0,0,71,46,0,0,0,0,67,6,0,0,6,6] >;

C3×C8.12D4 in GAP, Magma, Sage, TeX

C_3\times C_8._{12}D_4
% in TeX

G:=Group("C3xC8.12D4");
// GroupNames label

G:=SmallGroup(192,928);
// by ID

G=gap.SmallGroup(192,928);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,772,4204,172]);
// Polycyclic

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

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