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## G = C3×C23.32C23order 192 = 26·3

### Direct product of C3 and C23.32C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C23.32C23
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C4⋊C4 — Q8×C12 — C3×C23.32C23
 Lower central C1 — C2 — C3×C23.32C23
 Upper central C1 — C2×C6 — C3×C23.32C23

Generators and relations for C3×C23.32C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=d, f2=g2=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, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 290 in 266 conjugacy classes, 242 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×C12, C3×Q8, C22×C6, C42⋊C2, C4×Q8, C22×Q8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C23.32C23, C3×C42⋊C2, Q8×C12, Q8×C2×C6, C3×C23.32C23
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C24, C2×C12, C22×C6, C23×C4, 2- 1+4, C22×C12, C23×C6, C23.32C23, C23×C12, C3×2- 1+4, C3×C23.32C23

Smallest permutation representation of C3×C23.32C23
On 96 points
Generators in S96
(1 57 9)(2 58 10)(3 59 11)(4 60 12)(5 28 54)(6 25 55)(7 26 56)(8 27 53)(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 96)(6 8)(7 94)(9 11)(10 30)(12 32)(13 15)(14 34)(16 36)(17 19)(18 38)(20 40)(21 23)(22 42)(24 44)(25 27)(26 46)(28 48)(29 31)(33 35)(37 39)(41 43)(45 47)(49 51)(50 54)(52 56)(53 55)(57 59)(58 78)(60 80)(61 63)(62 82)(64 84)(65 67)(66 86)(68 88)(69 71)(70 90)(72 92)(73 75)(77 79)(81 83)(85 87)(89 91)(93 95)
(1 75)(2 76)(3 73)(4 74)(5 94)(6 95)(7 96)(8 93)(9 31)(10 32)(11 29)(12 30)(13 35)(14 36)(15 33)(16 34)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(25 47)(26 48)(27 45)(28 46)(49 55)(50 56)(51 53)(52 54)(57 79)(58 80)(59 77)(60 78)(61 83)(62 84)(63 81)(64 82)(65 87)(66 88)(67 85)(68 86)(69 91)(70 92)(71 89)(72 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 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 65)(2 66 76 88)(3 85 73 67)(4 68 74 86)(5 82 94 64)(6 61 95 83)(7 84 96 62)(8 63 93 81)(9 43 31 21)(10 22 32 44)(11 41 29 23)(12 24 30 42)(13 47 35 25)(14 26 36 48)(15 45 33 27)(16 28 34 46)(17 49 39 55)(18 56 40 50)(19 51 37 53)(20 54 38 52)(57 91 79 69)(58 70 80 92)(59 89 77 71)(60 72 78 90)
(1 35 75 13)(2 36 76 14)(3 33 73 15)(4 34 74 16)(5 42 94 24)(6 43 95 21)(7 44 96 22)(8 41 93 23)(9 83 31 61)(10 84 32 62)(11 81 29 63)(12 82 30 64)(17 57 39 79)(18 58 40 80)(19 59 37 77)(20 60 38 78)(25 87 47 65)(26 88 48 66)(27 85 45 67)(28 86 46 68)(49 69 55 91)(50 70 56 92)(51 71 53 89)(52 72 54 90)

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

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

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

102 conjugacy classes

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

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 2- 1+4 C3×2- 1+4 kernel C3×C23.32C23 C3×C42⋊C2 Q8×C12 Q8×C2×C6 C23.32C23 C6×Q8 C42⋊C2 C4×Q8 C22×Q8 C2×Q8 C6 C2 # reps 1 6 8 1 2 16 12 16 2 32 2 4

Matrix representation of C3×C23.32C23 in GL6(𝔽13)

 1 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 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 1 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
,
 8 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0

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

C3×C23.32C23 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{32}C_2^3
% in TeX

G:=Group("C3xC2^3.32C2^3");
// GroupNames label

G:=SmallGroup(192,1408);
// by ID

G=gap.SmallGroup(192,1408);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,344,555,268,1571]);
// Polycyclic

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

׿
×
𝔽