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## G = C6×2- 1+4order 192 = 26·3

### Direct product of C6 and 2- 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×2- 1+4
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C3×C4○D4 — C3×2- 1+4 — C6×2- 1+4
 Lower central C1 — C2 — C6×2- 1+4
 Upper central C1 — C2×C6 — C6×2- 1+4

Generators and relations for C6×2- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 834 in 794 conjugacy classes, 754 normal (8 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C3×D4, C3×Q8, C22×C6, C22×Q8, C2×C4○D4, 2- 1+4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2×2- 1+4, Q8×C2×C6, C6×C4○D4, C3×2- 1+4, C6×2- 1+4
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2- 1+4, C25, C23×C6, C2×2- 1+4, C3×2- 1+4, C24×C6, C6×2- 1+4

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

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 3A 3B 4A ··· 4T 6A ··· 6F 6G ··· 6Z 12A ··· 12AN order 1 2 2 2 2 ··· 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 ··· 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 type + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 2- 1+4 C3×2- 1+4 kernel C6×2- 1+4 Q8×C2×C6 C6×C4○D4 C3×2- 1+4 C2×2- 1+4 C22×Q8 C2×C4○D4 2- 1+4 C6 C2 # reps 1 5 10 16 2 10 20 32 2 4

Matrix representation of C6×2- 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 9 10 0 0 0 0 10 4 0 0 9 10 0 0 0 0 10 4 0 0
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 10 4 0 0 0 0 4 3 0 0 0 0 0 0 3 9 0 0 0 0 9 10

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,10,0,0,0,0,10,4,0,0,9,10,0,0,0,0,10,4,0,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,4,0,0,0,0,4,3,0,0,0,0,0,0,3,9,0,0,0,0,9,10] >;

C6×2- 1+4 in GAP, Magma, Sage, TeX

C_6\times 2_-^{1+4}
% in TeX

G:=Group("C6xES-(2,2)");
// GroupNames label

G:=SmallGroup(192,1535);
// by ID

G=gap.SmallGroup(192,1535);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-2,1373,680,1059,520,2915]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations

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