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## G = C3×D8.C4order 192 = 26·3

### Direct product of C3 and D8.C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×D8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C24 — C3×C8.C4 — C3×D8.C4
 Lower central C1 — C2 — C4 — C8 — C3×D8.C4
 Upper central C1 — C12 — C2×C12 — C2×C24 — C3×D8.C4

Generators and relations for C3×D8.C4
G = < a,b,c,d | a3=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E 12F 12G 12H 16A ··· 16H 24A ··· 24H 24I 24J 24K 24L 48A ··· 48P order 1 2 2 2 3 3 4 4 4 4 6 6 6 6 6 6 8 8 8 8 8 8 12 12 12 12 12 12 12 12 16 ··· 16 24 ··· 24 24 24 24 24 48 ··· 48 size 1 1 2 8 1 1 1 1 2 8 1 1 2 2 8 8 2 2 2 2 8 8 1 1 1 1 2 2 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 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 D8 SD16 C3×D4 C3×D4 C3×D8 C3×SD16 D8.C4 C3×D8.C4 kernel C3×D8.C4 C3×C8.C4 C2×C48 C3×C4○D8 D8.C4 C3×D8 C3×Q16 C8.C4 C2×C16 C4○D8 D8 Q16 C24 C2×C12 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 2 4 4 8 16

Matrix representation of C3×D8.C4 in GL2(𝔽97) generated by

 61 0 0 61
,
 7 90 7 7
,
 7 7 7 90
,
 39 94 94 58
G:=sub<GL(2,GF(97))| [61,0,0,61],[7,7,90,7],[7,7,7,90],[39,94,94,58] >;

C3×D8.C4 in GAP, Magma, Sage, TeX

C_3\times D_8.C_4
% in TeX

G:=Group("C3xD8.C4");
// GroupNames label

G:=SmallGroup(192,165);
// by ID

G=gap.SmallGroup(192,165);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,850,360,172,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^2=1,d^4=b^4,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^5*c>;
// generators/relations

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