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## G = C3×C24.3C22order 192 = 26·3

### Direct product of C3 and C24.3C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C24.3C22
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C6×C22⋊C4 — C3×C24.3C22
 Lower central C1 — C22 — C3×C24.3C22
 Upper central C1 — C22×C6 — C3×C24.3C22

Generators and relations for C3×C24.3C22
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=1, f2=e, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 466 in 258 conjugacy classes, 106 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C6×D4, C23×C6, C24.3C22, C2×C4×C12, C6×C22⋊C4, C6×C4⋊C4, D4×C2×C6, C3×C24.3C22
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C41D4, C3×C22⋊C4, C22×C12, C6×D4, C3×C4○D4, C24.3C22, C6×C22⋊C4, D4×C12, C3×C4⋊D4, C3×C4.4D4, C3×C41D4, C3×C24.3C22

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

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 4A ··· 4L 4M 4N 4O 4P 6A ··· 6N 6O ··· 6V 12A ··· 12X 12Y ··· 12AF order 1 2 ··· 2 2 2 2 2 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 4 4 4 4 1 1 2 ··· 2 4 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D4 C4○D4 C3×D4 C3×C4○D4 kernel C3×C24.3C22 C2×C4×C12 C6×C22⋊C4 C6×C4⋊C4 D4×C2×C6 C24.3C22 C6×D4 C2×C42 C2×C22⋊C4 C2×C4⋊C4 C22×D4 C2×D4 C2×C12 C2×C6 C2×C4 C22 # reps 1 1 4 1 1 2 8 2 8 2 2 16 8 4 16 8

Matrix representation of C3×C24.3C22 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 2 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 12 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 0 0 0 0 0 0 12
,
 12 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 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 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0 12
,
 1 12 0 0 0 0 2 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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],[1,2,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,8,8,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,12],[1,2,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×C24.3C22 in GAP, Magma, Sage, TeX

C_3\times C_2^4._3C_2^2
% in TeX

G:=Group("C3xC2^4.3C2^2");
// GroupNames label

G:=SmallGroup(192,823);
// by ID

G=gap.SmallGroup(192,823);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,848,1094,268]);
// Polycyclic

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

׿
×
𝔽