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## G = C3×C22.26C24order 192 = 26·3

### Direct product of C3 and C22.26C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.26C24
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊D4 — C3×C22.26C24
 Lower central C1 — C22 — C3×C22.26C24
 Upper central C1 — C2×C12 — C3×C22.26C24

Generators and relations for C3×C22.26C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=1, f2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 466 in 310 conjugacy classes, 170 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C2×C42, C4×D4, C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C22.26C24, C2×C4×C12, D4×C12, C3×C4⋊D4, C3×C4.4D4, C3×C41D4, C3×C4⋊Q8, C6×C4○D4, C3×C22.26C24
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C24, C3×D4, C22×C6, C22×D4, C2×C4○D4, C6×D4, C3×C4○D4, C23×C6, C22.26C24, D4×C2×C6, C6×C4○D4, C3×C22.26C24

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 6A ··· 6F 6G 6H 6I 6J 6K ··· 6R 12A ··· 12H 12I ··· 12AB 12AC ··· 12AJ order 1 2 2 2 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 kernel C3×C22.26C24 C2×C4×C12 D4×C12 C3×C4⋊D4 C3×C4.4D4 C3×C4⋊1D4 C3×C4⋊Q8 C6×C4○D4 C22.26C24 C2×C42 C4×D4 C4⋊D4 C4.4D4 C4⋊1D4 C4⋊Q8 C2×C4○D4 C2×C12 C12 C2×C4 C4 # reps 1 1 4 4 2 1 1 2 2 2 8 8 4 2 2 4 4 8 8 16

Matrix representation of C3×C22.26C24 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 12 0 0 0 0 12 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
,
 12 0 0 0 11 1 0 0 0 0 12 0 0 0 1 1
,
 5 8 0 0 10 8 0 0 0 0 12 11 0 0 0 1
,
 1 12 0 0 2 12 0 0 0 0 12 0 0 0 0 12
,
 8 0 0 0 0 8 0 0 0 0 5 0 0 0 0 5
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,11,0,0,0,1,0,0,0,0,12,1,0,0,0,1],[5,10,0,0,8,8,0,0,0,0,12,0,0,0,11,1],[1,2,0,0,12,12,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,5,0,0,0,0,5] >;

C3×C22.26C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{26}C_2^4
% in TeX

G:=Group("C3xC2^2.26C2^4");
// GroupNames label

G:=SmallGroup(192,1421);
// by ID

G=gap.SmallGroup(192,1421);
# by ID

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

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

׿
×
𝔽