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## G = C3×C23.33C23order 192 = 26·3

### Direct product of C3 and C23.33C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C23.33C23
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — D4×C12 — C3×C23.33C23
 Lower central C1 — C2 — C3×C23.33C23
 Upper central C1 — C2×C6 — C3×C23.33C23

Generators and relations for C3×C23.33C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=f2=1, e2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf=bc=cb, bd=db, be=eb, bg=gb, cd=dc, geg-1=ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe >

Subgroups: 370 in 294 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C23.33C23, C6×C4⋊C4, C3×C42⋊C2, D4×C12, Q8×C12, C6×C4○D4, C3×C23.33C23
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, 2- 1+4, C22×C12, C23×C6, C23.33C23, C23×C12, C3×2+ 1+4, C3×2- 1+4, C3×C23.33C23

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A ··· 4X 6A ··· 6F 6G ··· 6R 12A ··· 12AV order 1 2 2 2 2 ··· 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 ··· 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 2+ 1+4 2- 1+4 C3×2+ 1+4 C3×2- 1+4 kernel C3×C23.33C23 C6×C4⋊C4 C3×C42⋊C2 D4×C12 Q8×C12 C6×C4○D4 C23.33C23 C3×C4○D4 C2×C4⋊C4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C6 C6 C2 C2 # reps 1 3 3 6 2 1 2 16 6 6 12 4 2 32 1 1 2 2

Matrix representation of C3×C23.33C23 in GL5(𝔽13)

 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 12 1 12 0 0 2 12 0 11 0 2 11 1 11 0 11 1 12 1
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 8 0 0 0 0 0 12 0 12 0 0 11 1 0 2 0 0 0 1 0 0 0 0 12 12
,
 12 0 0 0 0 0 12 0 12 0 0 0 12 0 11 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 0 12 0 0 0 1 0 2 0 2 0 1 0 0 0 12 0 12

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

C3×C23.33C23 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{33}C_2^3
% in TeX

G:=Group("C3xC2^3.33C2^3");
// GroupNames label

G:=SmallGroup(192,1409);
// by ID

G=gap.SmallGroup(192,1409);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,555,1571,192]);
// Polycyclic

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

׿
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