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## G = C3×C22.31C24order 192 = 26·3

### Direct product of C3 and C22.31C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.31C24
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊D4 — C3×C22.31C24
 Lower central C1 — C22 — C3×C22.31C24
 Upper central C1 — C2×C6 — C3×C22.31C24

Generators and relations for C3×C22.31C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=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: 466 in 294 conjugacy classes, 162 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×C4○D4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C22.31C24, C6×C4⋊C4, C3×C4⋊D4, C3×C22⋊Q8, C6×C4○D4, C3×C22.31C24
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, 2+ 1+4, 2- 1+4, C6×D4, C23×C6, C22.31C24, D4×C2×C6, C3×2+ 1+4, C3×2- 1+4, C3×C22.31C24

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G 6H 6I 6J 6K ··· 6R 12A ··· 12H 12I ··· 12X order 1 2 2 2 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 4 4 4 4 1 1 2 2 2 2 4 ··· 4 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 C3×D4 2+ 1+4 2- 1+4 C3×2+ 1+4 C3×2- 1+4 kernel C3×C22.31C24 C6×C4⋊C4 C3×C4⋊D4 C3×C22⋊Q8 C6×C4○D4 C22.31C24 C2×C4⋊C4 C4⋊D4 C22⋊Q8 C2×C4○D4 C2×C12 C2×C4 C6 C6 C2 C2 # reps 1 1 8 4 2 2 2 16 8 4 4 8 1 1 2 2

Matrix representation of C3×C22.31C24 in GL6(𝔽13)

 9 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 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
,
 1 0 0 0 0 0 11 12 0 0 0 0 0 0 9 9 0 0 0 0 7 4 0 0 0 0 0 2 7 11 0 0 7 11 11 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 2 0 0 12 0 12 1 0 0 1 12 0 12 0 0 0 0 0 1
,
 1 1 0 0 0 0 0 12 0 0 0 0 0 0 1 0 2 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 1 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 2 0 0 0 0 12 1 0 0 0 0 0 12 0 1 0 0 12 1 12 0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[1,11,0,0,0,0,0,12,0,0,0,0,0,0,9,7,0,7,0,0,9,4,2,11,0,0,0,0,7,11,0,0,0,0,11,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,1,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,2,1,12,1],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,1,12,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,12,0,0,2,1,12,1,0,0,0,0,0,12,0,0,0,0,1,0] >;

C3×C22.31C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{31}C_2^4
% in TeX

G:=Group("C3xC2^2.31C2^4");
// GroupNames label

G:=SmallGroup(192,1426);
// by ID

G=gap.SmallGroup(192,1426);
# by ID

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

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