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## G = C2×C6×M4(2)  order 192 = 26·3

### Direct product of C2×C6 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2×C6×M4(2)
 Chief series C1 — C2 — C4 — C12 — C24 — C3×M4(2) — C6×M4(2) — C2×C6×M4(2)
 Lower central C1 — C2 — C2×C6×M4(2)
 Upper central C1 — C22×C12 — C2×C6×M4(2)

Generators and relations for C2×C6×M4(2)
G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 338 in 298 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C22×C4, C24, C24, C2×C12, C22×C6, C22×C6, C22×C6, C22×C8, C2×M4(2), C23×C4, C2×C24, C3×M4(2), C22×C12, C22×C12, C23×C6, C22×M4(2), C22×C24, C6×M4(2), C23×C12, C2×C6×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, M4(2), C22×C4, C24, C2×C12, C22×C6, C2×M4(2), C23×C4, C3×M4(2), C22×C12, C23×C6, C22×M4(2), C6×M4(2), C23×C12, C2×C6×M4(2)

Smallest permutation representation of C2×C6×M4(2)
On 96 points
Generators in S96
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 87)(26 88)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)(41 93)(42 94)(43 95)(44 96)(45 89)(46 90)(47 91)(48 92)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)
(1 55 25 91 19 78)(2 56 26 92 20 79)(3 49 27 93 21 80)(4 50 28 94 22 73)(5 51 29 95 23 74)(6 52 30 96 24 75)(7 53 31 89 17 76)(8 54 32 90 18 77)(9 65 81 41 64 37)(10 66 82 42 57 38)(11 67 83 43 58 39)(12 68 84 44 59 40)(13 69 85 45 60 33)(14 70 86 46 61 34)(15 71 87 47 62 35)(16 72 88 48 63 36)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)(82 86)(84 88)(90 94)(92 96)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 4A ··· 4H 4I 4J 4K 4L 6A ··· 6N 6O ··· 6V 8A ··· 8P 12A ··· 12P 12Q ··· 12X 24A ··· 24AF order 1 2 ··· 2 2 2 2 2 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 ··· 1 2 2 2 2 1 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 M4(2) C3×M4(2) kernel C2×C6×M4(2) C22×C24 C6×M4(2) C23×C12 C22×M4(2) C22×C12 C23×C6 C22×C8 C2×M4(2) C23×C4 C22×C4 C24 C2×C6 C22 # reps 1 2 12 1 2 14 2 4 24 2 28 4 8 16

Matrix representation of C2×C6×M4(2) in GL4(𝔽73) generated by

 72 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 9 0 0 0 0 65 0 0 0 0 65
,
 1 0 0 0 0 72 0 0 0 0 19 18 0 0 19 54
,
 72 0 0 0 0 72 0 0 0 0 1 71 0 0 0 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,9,0,0,0,0,65,0,0,0,0,65],[1,0,0,0,0,72,0,0,0,0,19,19,0,0,18,54],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,71,72] >;

C2×C6×M4(2) in GAP, Magma, Sage, TeX

C_2\times C_6\times M_4(2)
% in TeX

G:=Group("C2xC6xM4(2)");
// GroupNames label

G:=SmallGroup(192,1455);
// by ID

G=gap.SmallGroup(192,1455);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,1373,124]);
// 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^5>;
// generators/relations

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