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## G = C3×C22.49C24order 192 = 26·3

### Direct product of C3 and C22.49C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.49C24
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C3×C22⋊C4 — C3×C4.4D4 — C3×C22.49C24
 Lower central C1 — C22 — C3×C22.49C24
 Upper central C1 — C2×C6 — C3×C22.49C24

Generators and relations for C3×C22.49C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 362 in 236 conjugacy classes, 150 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C22.49C24, C3×C42⋊C2, D4×C12, C3×C4⋊D4, C3×C4.4D4, C3×C4⋊Q8, C3×C22.49C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C2×C4○D4, 2+ 1+4, C3×C4○D4, C23×C6, C22.49C24, C6×C4○D4, C3×2+ 1+4, C3×C22.49C24

Smallest permutation representation of C3×C22.49C24
On 96 points
Generators in S96
(1 57 9)(2 58 10)(3 59 11)(4 60 12)(5 28 54)(6 25 55)(7 26 56)(8 27 53)(13 17 61)(14 18 62)(15 19 63)(16 20 64)(21 65 69)(22 66 70)(23 67 71)(24 68 72)(29 73 77)(30 74 78)(31 75 79)(32 76 80)(33 37 81)(34 38 82)(35 39 83)(36 40 84)(41 85 89)(42 86 90)(43 87 91)(44 88 92)(45 51 93)(46 52 94)(47 49 95)(48 50 96)
(1 75)(2 76)(3 73)(4 74)(5 94)(6 95)(7 96)(8 93)(9 31)(10 32)(11 29)(12 30)(13 35)(14 36)(15 33)(16 34)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(25 47)(26 48)(27 45)(28 46)(49 55)(50 56)(51 53)(52 54)(57 79)(58 80)(59 77)(60 78)(61 83)(62 84)(63 81)(64 82)(65 87)(66 88)(67 85)(68 86)(69 91)(70 92)(71 89)(72 90)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)
(1 28)(2 47)(3 26)(4 45)(5 9)(6 32)(7 11)(8 30)(10 95)(12 93)(13 68)(14 87)(15 66)(16 85)(17 72)(18 91)(19 70)(20 89)(21 84)(22 63)(23 82)(24 61)(25 76)(27 74)(29 96)(31 94)(33 88)(34 67)(35 86)(36 65)(37 92)(38 71)(39 90)(40 69)(41 64)(42 83)(43 62)(44 81)(46 75)(48 73)(49 58)(50 77)(51 60)(52 79)(53 78)(54 57)(55 80)(56 59)
(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 87 75 65)(2 88 76 66)(3 85 73 67)(4 86 74 68)(5 64 94 82)(6 61 95 83)(7 62 96 84)(8 63 93 81)(9 43 31 21)(10 44 32 22)(11 41 29 23)(12 42 30 24)(13 47 35 25)(14 48 36 26)(15 45 33 27)(16 46 34 28)(17 49 39 55)(18 50 40 56)(19 51 37 53)(20 52 38 54)(57 91 79 69)(58 92 80 70)(59 89 77 71)(60 90 78 72)
(1 35 75 13)(2 14 76 36)(3 33 73 15)(4 16 74 34)(5 42 94 24)(6 21 95 43)(7 44 96 22)(8 23 93 41)(9 83 31 61)(10 62 32 84)(11 81 29 63)(12 64 30 82)(17 57 39 79)(18 80 40 58)(19 59 37 77)(20 78 38 60)(25 65 47 87)(26 88 48 66)(27 67 45 85)(28 86 46 68)(49 91 55 69)(50 70 56 92)(51 89 53 71)(52 72 54 90)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4L 4M ··· 4Q 6A ··· 6F 6G ··· 6N 12A ··· 12X 12Y ··· 12AH order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 4 4 4 4 1 1 2 ··· 2 4 ··· 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C4○D4 C3×C4○D4 2+ 1+4 C3×2+ 1+4 kernel C3×C22.49C24 C3×C42⋊C2 D4×C12 C3×C4⋊D4 C3×C4.4D4 C3×C4⋊Q8 C22.49C24 C42⋊C2 C4×D4 C4⋊D4 C4.4D4 C4⋊Q8 C12 C4 C6 C2 # reps 1 4 2 4 4 1 2 8 4 8 8 2 8 16 1 2

Matrix representation of C3×C22.49C24 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 0 8 0 0 5 0 0 0 0 0 8 10 0 0 8 5
,
 8 0 0 0 0 8 0 0 0 0 12 0 0 0 12 1
,
 0 1 0 0 1 0 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 12 2 0 0 12 1
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,5,0,0,8,0,0,0,0,0,8,8,0,0,10,5],[8,0,0,0,0,8,0,0,0,0,12,12,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,2,1] >;

C3×C22.49C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{49}C_2^4
% in TeX

G:=Group("C3xC2^2.49C2^4");
// GroupNames label

G:=SmallGroup(192,1444);
// by ID

G=gap.SmallGroup(192,1444);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,268,794,192]);
// Polycyclic

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

׿
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