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## G = C3×C22.53C24order 192 = 26·3

### Direct product of C3 and C22.53C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.53C24
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — C3×C4.4D4 — C3×C22.53C24
 Lower central C1 — C22 — C3×C22.53C24
 Upper central C1 — C2×C6 — C3×C22.53C24

Generators and relations for C3×C22.53C24
G = < a,b,c,d,e,f,g | a3=b2=c2=1, d2=f2=c, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, 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, C4×D4, C4×Q8, C22.D4, C4.4D4, C41D4, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C22.53C24, D4×C12, Q8×C12, C3×C22.D4, C3×C4.4D4, C3×C41D4, C3×C22.53C24
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, C22.53C24, C6×C4○D4, C3×2+ 1+4, C3×C22.53C24

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

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

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

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

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×C22.53C24 D4×C12 Q8×C12 C3×C22.D4 C3×C4.4D4 C3×C4⋊1D4 C22.53C24 C4×D4 C4×Q8 C22.D4 C4.4D4 C4⋊1D4 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×C22.53C24 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 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
,
 5 10 0 0 0 8 0 0 0 0 5 3 0 0 5 8
,
 8 0 0 0 0 8 0 0 0 0 5 0 0 0 5 8
,
 12 11 0 0 1 1 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 11 0 0 1 12
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[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],[5,0,0,0,10,8,0,0,0,0,5,5,0,0,3,8],[8,0,0,0,0,8,0,0,0,0,5,5,0,0,0,8],[12,1,0,0,11,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,1,0,0,11,12] >;

C3×C22.53C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{53}C_2^4
% in TeX

G:=Group("C3xC2^2.53C2^4");
// GroupNames label

G:=SmallGroup(192,1448);
// by ID

G=gap.SmallGroup(192,1448);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,1016,2102,520,794,192]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=1,d^2=f^2=c,e^2=c*b=b*c,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,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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