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## G = C3×C23.4Q8order 192 = 26·3

### Direct product of C3 and C23.4Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×C23.4Q8
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C6×C4⋊C4 — C3×C23.4Q8
 Lower central C1 — C23 — C3×C23.4Q8
 Upper central C1 — C22×C6 — C3×C23.4Q8

Generators and relations for C3×C23.4Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 330 in 186 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C23×C6, C23.4Q8, C3×C2.C42, C6×C22⋊C4, C6×C4⋊C4, C3×C23.4Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C22.D4, C41D4, C6×D4, C6×Q8, C3×C4○D4, C23.4Q8, C3×C22⋊Q8, C3×C22.D4, C3×C41D4, C3×C23.4Q8

Smallest permutation representation of C3×C23.4Q8
On 96 points
Generators in S96
(1 33 39)(2 34 40)(3 35 37)(4 36 38)(5 94 92)(6 95 89)(7 96 90)(8 93 91)(9 17 15)(10 18 16)(11 19 13)(12 20 14)(21 49 55)(22 50 56)(23 51 53)(24 52 54)(25 68 31)(26 65 32)(27 66 29)(28 67 30)(41 73 47)(42 74 48)(43 75 45)(44 76 46)(57 78 71)(58 79 72)(59 80 69)(60 77 70)(61 88 82)(62 85 83)(63 86 84)(64 87 81)
(1 9)(2 42)(3 11)(4 44)(5 88)(6 8)(7 86)(10 24)(12 22)(13 37)(14 56)(15 39)(16 54)(17 33)(18 52)(19 35)(20 50)(21 43)(23 41)(25 72)(26 28)(27 70)(29 77)(30 32)(31 79)(34 74)(36 76)(38 46)(40 48)(45 55)(47 53)(49 75)(51 73)(57 59)(58 68)(60 66)(61 92)(62 64)(63 90)(65 67)(69 71)(78 80)(81 83)(82 94)(84 96)(85 87)(89 91)(93 95)
(1 11)(2 12)(3 9)(4 10)(5 31)(6 32)(7 29)(8 30)(13 39)(14 40)(15 37)(16 38)(17 35)(18 36)(19 33)(20 34)(21 41)(22 42)(23 43)(24 44)(25 94)(26 95)(27 96)(28 93)(45 53)(46 54)(47 55)(48 56)(49 73)(50 74)(51 75)(52 76)(57 64)(58 61)(59 62)(60 63)(65 89)(66 90)(67 91)(68 92)(69 83)(70 84)(71 81)(72 82)(77 86)(78 87)(79 88)(80 85)
(1 23)(2 24)(3 21)(4 22)(5 86)(6 87)(7 88)(8 85)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 73)(18 74)(19 75)(20 76)(25 70)(26 71)(27 72)(28 69)(29 79)(30 80)(31 77)(32 78)(33 51)(34 52)(35 49)(36 50)(37 55)(38 56)(39 53)(40 54)(57 65)(58 66)(59 67)(60 68)(61 90)(62 91)(63 92)(64 89)(81 95)(82 96)(83 93)(84 94)
(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 59 9 64)(2 61 10 60)(3 57 11 62)(4 63 12 58)(5 76 29 50)(6 51 30 73)(7 74 31 52)(8 49 32 75)(13 83 37 71)(14 72 38 84)(15 81 39 69)(16 70 40 82)(17 87 33 80)(18 77 34 88)(19 85 35 78)(20 79 36 86)(21 65 43 91)(22 92 44 66)(23 67 41 89)(24 90 42 68)(25 54 96 48)(26 45 93 55)(27 56 94 46)(28 47 95 53)

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

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

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

66 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3A 3B 4A ··· 4L 6A ··· 6N 6O 6P 6Q 6R 12A ··· 12X order 1 2 ··· 2 2 2 3 3 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 1 1 4 ··· 4 1 ··· 1 4 4 4 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 kernel C3×C23.4Q8 C3×C2.C42 C6×C22⋊C4 C6×C4⋊C4 C23.4Q8 C2.C42 C2×C22⋊C4 C2×C4⋊C4 C2×C12 C22×C6 C2×C6 C2×C4 C23 C22 # reps 1 1 3 3 2 2 6 6 6 2 6 12 4 12

Matrix representation of C3×C23.4Q8 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 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
,
 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
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 0 5 0

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

C3×C23.4Q8 in GAP, Magma, Sage, TeX

C_3\times C_2^3._4Q_8
% in TeX

G:=Group("C3xC2^3.4Q8");
// GroupNames label

G:=SmallGroup(192,832);
// by ID

G=gap.SmallGroup(192,832);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,512,1094,1059]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations

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