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## G = C3×C22.34C24order 192 = 26·3

### Direct product of C3 and C22.34C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.34C24
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C3×C22⋊C4 — C3×C4⋊D4 — C3×C22.34C24
 Lower central C1 — C22 — C3×C22.34C24
 Upper central C1 — C2×C6 — C3×C22.34C24

Generators and relations for C3×C22.34C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=f2=1, e2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 402 in 240 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C41D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C22.34C24, C3×C42⋊C2, D4×C12, C3×C4⋊D4, C3×C22.D4, C3×C42.C2, C3×C41D4, C3×C22.34C24
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, C22.34C24, C6×C4○D4, C3×2+ 1+4, C3×C22.34C24

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2H 3A 3B 4A ··· 4F 4G ··· 4M 6A ··· 6F 6G ··· 6P 12A ··· 12L 12M ··· 12Z order 1 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 1 1 2 ··· 2 4 ··· 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C4○D4 C3×C4○D4 2+ 1+4 C3×2+ 1+4 kernel C3×C22.34C24 C3×C42⋊C2 D4×C12 C3×C4⋊D4 C3×C22.D4 C3×C42.C2 C3×C4⋊1D4 C22.34C24 C42⋊C2 C4×D4 C4⋊D4 C22.D4 C42.C2 C4⋊1D4 C12 C4 C6 C2 # reps 1 1 2 6 4 1 1 2 2 4 12 8 2 2 4 8 2 4

Matrix representation of C3×C22.34C24 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 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 12
,
 12 0 0 0 0 0 0 12 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
,
 5 8 0 0 0 0 10 8 0 0 0 0 0 0 1 0 0 12 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 12 0 12 1 0 0 12 12 0 1 0 0 11 0 0 1
,
 1 0 0 0 0 0 2 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 11 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1 12 0 0 12 0 1 0 0 0 0 0 1 12 0 0 0 0 2 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,10,0,0,0,0,8,8,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,12,0,12,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,12,12,11,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,1,1,1],[1,2,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,12,12,12,11,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,1,1,1,2,0,0,12,0,12,12] >;

C3×C22.34C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{34}C_2^4
% in TeX

G:=Group("C3xC2^2.34C2^4");
// GroupNames label

G:=SmallGroup(192,1429);
// by ID

G=gap.SmallGroup(192,1429);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,555,1571,192]);
// Polycyclic

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

׿
×
𝔽