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## G = C3×C23.41C23order 192 = 26·3

### Direct product of C3 and C23.41C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23.41C23
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C4⋊C4 — C3×C4⋊Q8 — C3×C23.41C23
 Lower central C1 — C22 — C3×C23.41C23
 Upper central C1 — C2×C6 — C3×C23.41C23

Generators and relations for C3×C23.41C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=g2=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, geg-1=de=ed, df=fd, dg=gd >

Subgroups: 274 in 206 conjugacy classes, 162 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C12, C2×C12, C3×Q8, C22×C6, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×Q8, C23.41C23, C6×C4⋊C4, C3×C42⋊C2, C3×C22⋊Q8, C3×C42.C2, C3×C4⋊Q8, C3×C23.41C23
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C24, C3×Q8, C22×C6, C22×Q8, 2+ 1+4, 2- 1+4, C6×Q8, C23×C6, C23.41C23, Q8×C2×C6, C3×2+ 1+4, C3×2- 1+4, C3×C23.41C23

Smallest permutation representation of C3×C23.41C23
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 3)(2 74)(4 76)(5 50)(6 8)(7 52)(9 11)(10 32)(12 30)(13 15)(14 36)(16 34)(17 19)(18 40)(20 38)(21 23)(22 44)(24 42)(25 27)(26 48)(28 46)(29 31)(33 35)(37 39)(41 43)(45 47)(49 51)(53 55)(54 96)(56 94)(57 59)(58 80)(60 78)(61 63)(62 84)(64 82)(65 67)(66 88)(68 86)(69 71)(70 92)(72 90)(73 75)(77 79)(81 83)(85 87)(89 91)(93 95)
(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 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 87 75 67)(2 68 76 88)(3 85 73 65)(4 66 74 86)(5 38 52 18)(6 19 49 39)(7 40 50 20)(8 17 51 37)(9 41 29 21)(10 22 30 42)(11 43 31 23)(12 24 32 44)(13 45 33 25)(14 26 34 46)(15 47 35 27)(16 28 36 48)(53 61 93 81)(54 82 94 62)(55 63 95 83)(56 84 96 64)(57 89 77 69)(58 70 78 90)(59 91 79 71)(60 72 80 92)
(1 13 3 15)(2 16 4 14)(5 92 7 90)(6 91 8 89)(9 63 11 61)(10 62 12 64)(17 57 19 59)(18 60 20 58)(21 95 23 93)(22 94 24 96)(25 85 27 87)(26 88 28 86)(29 83 31 81)(30 82 32 84)(33 73 35 75)(34 76 36 74)(37 77 39 79)(38 80 40 78)(41 55 43 53)(42 54 44 56)(45 65 47 67)(46 68 48 66)(49 71 51 69)(50 70 52 72)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E ··· 4P 6A ··· 6F 6G 6H 6I 6J 12A ··· 12H 12I ··· 12AF order 1 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 1 1 2 2 2 2 4 ··· 4 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + + - + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 Q8 C3×Q8 2+ 1+4 2- 1+4 C3×2+ 1+4 C3×2- 1+4 kernel C3×C23.41C23 C6×C4⋊C4 C3×C42⋊C2 C3×C22⋊Q8 C3×C42.C2 C3×C4⋊Q8 C23.41C23 C2×C4⋊C4 C42⋊C2 C22⋊Q8 C42.C2 C4⋊Q8 C2×C12 C2×C4 C6 C6 C2 C2 # reps 1 1 2 4 4 4 2 2 4 8 8 8 4 8 1 1 2 2

Matrix representation of C3×C23.41C23 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 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 9 12 1 0 0 0 8 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 4 1 11 0 0 0 5 0 0 11 0 0 11 2 9 12 0 0 10 3 8 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 6 0 5 0 0 0 0 0 0 8
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 2 2 0 12 0 0 11 2 1 0

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,9,8,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,4,5,11,10,0,0,1,0,2,3,0,0,11,0,9,8,0,0,0,11,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,6,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,2,11,0,0,1,0,2,2,0,0,0,0,0,1,0,0,0,0,12,0] >;

C3×C23.41C23 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{41}C_2^3
% in TeX

G:=Group("C3xC2^3.41C2^3");
// GroupNames label

G:=SmallGroup(192,1433);
// by ID

G=gap.SmallGroup(192,1433);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,344,2102,555,520,1571]);
// Polycyclic

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

׿
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