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## G = C2×C6×D8order 192 = 26·3

### Direct product of C2×C6 and D8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C6×D8
G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 658 in 338 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, D4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C24, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, C22×C8, C2×D8, C22×D4, C2×C24, C3×D8, C22×C12, C6×D4, C6×D4, C23×C6, C22×D8, C22×C24, C6×D8, D4×C2×C6, C2×C6×D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C24, C3×D4, C22×C6, C2×D8, C22×D4, C3×D8, C6×D4, C23×C6, C22×D8, C6×D8, D4×C2×C6, C2×C6×D8

Smallest permutation representation of C2×C6×D8
On 96 points
Generators in S96
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 72)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(41 76)(42 77)(43 78)(44 79)(45 80)(46 73)(47 74)(48 75)(49 87)(50 88)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 95)(58 96)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)
(1 28 53 95 22 75)(2 29 54 96 23 76)(3 30 55 89 24 77)(4 31 56 90 17 78)(5 32 49 91 18 79)(6 25 50 92 19 80)(7 26 51 93 20 73)(8 27 52 94 21 74)(9 66 82 64 38 47)(10 67 83 57 39 48)(11 68 84 58 40 41)(12 69 85 59 33 42)(13 70 86 60 34 43)(14 71 87 61 35 44)(15 72 88 62 36 45)(16 65 81 63 37 46)
(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 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(41 46)(42 45)(43 44)(47 48)(49 56)(50 55)(51 54)(52 53)(57 64)(58 63)(59 62)(60 61)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)

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

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

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

84 conjugacy classes

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

84 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 C2×C6×D8 C22×C24 C6×D8 D4×C2×C6 C22×D8 C22×C8 C2×D8 C22×D4 C2×C12 C22×C6 C2×C6 C2×C4 C23 C22 # reps 1 1 12 2 2 2 24 4 3 1 8 6 2 16

Matrix representation of C2×C6×D8 in GL7(𝔽73)

 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 0 32 38 0 0 0 0 0 71 41 0 0 0 0 0 0 0 72 2 0 0 0 0 0 72 1 0 0 0 0 0 0 0 57 16 0 0 0 0 0 57 57
,
 1 0 0 0 0 0 0 0 41 36 0 0 0 0 0 2 32 0 0 0 0 0 0 0 72 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 57 16 0 0 0 0 0 16 16

G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,32,71,0,0,0,0,0,38,41,0,0,0,0,0,0,0,72,72,0,0,0,0,0,2,1,0,0,0,0,0,0,0,57,57,0,0,0,0,0,16,57],[1,0,0,0,0,0,0,0,41,2,0,0,0,0,0,36,32,0,0,0,0,0,0,0,72,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,57,16,0,0,0,0,0,16,16] >;

C2×C6×D8 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_8
% in TeX

G:=Group("C2xC6xD8");
// GroupNames label

G:=SmallGroup(192,1458);
// by ID

G=gap.SmallGroup(192,1458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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