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## G = C3×C23.38D4order 192 = 26·3

### Direct product of C3 and C23.38D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C23.38D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×Q8⋊C4 — C3×C23.38D4
 Lower central C1 — C2 — C4 — C3×C23.38D4
 Upper central C1 — C2×C6 — C22×C12 — C3×C23.38D4

Generators and relations for C3×C23.38D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 226 in 150 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×C6, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C22×C12, C22×C12, C6×Q8, C6×Q8, C23.38D4, C3×Q8⋊C4, C3×C42⋊C2, C6×M4(2), Q8×C2×C6, C3×C23.38D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C8.C22, C3×C22⋊C4, C22×C12, C6×D4, C23.38D4, C6×C22⋊C4, C3×C8.C22, C3×C23.38D4

Smallest permutation representation of C3×C23.38D4
On 96 points
Generators in S96
(1 79 95)(2 80 96)(3 73 89)(4 74 90)(5 75 91)(6 76 92)(7 77 93)(8 78 94)(9 65 81)(10 66 82)(11 67 83)(12 68 84)(13 69 85)(14 70 86)(15 71 87)(16 72 88)(17 51 30)(18 52 31)(19 53 32)(20 54 25)(21 55 26)(22 56 27)(23 49 28)(24 50 29)(33 45 57)(34 46 58)(35 47 59)(36 48 60)(37 41 61)(38 42 62)(39 43 63)(40 44 64)
(1 21)(2 18)(3 23)(4 20)(5 17)(6 22)(7 19)(8 24)(9 38)(10 35)(11 40)(12 37)(13 34)(14 39)(15 36)(16 33)(25 90)(26 95)(27 92)(28 89)(29 94)(30 91)(31 96)(32 93)(41 68)(42 65)(43 70)(44 67)(45 72)(46 69)(47 66)(48 71)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)(57 88)(58 85)(59 82)(60 87)(61 84)(62 81)(63 86)(64 83)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(25 94)(26 95)(27 96)(28 89)(29 90)(30 91)(31 92)(32 93)(41 72)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 84)(58 85)(59 86)(60 87)(61 88)(62 81)(63 82)(64 83)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 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 43 21 66)(2 69 22 46)(3 41 23 72)(4 67 24 44)(5 47 17 70)(6 65 18 42)(7 45 19 68)(8 71 20 48)(9 31 38 92)(10 95 39 26)(11 29 40 90)(12 93 33 32)(13 27 34 96)(14 91 35 30)(15 25 36 94)(16 89 37 28)(49 88 73 61)(50 64 74 83)(51 86 75 59)(52 62 76 81)(53 84 77 57)(54 60 78 87)(55 82 79 63)(56 58 80 85)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A ··· 12H 12I ··· 12X 24A ··· 24H order 1 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 1 1 2 2 2 2 4 ··· 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D4 D4 C3×D4 C3×D4 C8.C22 C3×C8.C22 kernel C3×C23.38D4 C3×Q8⋊C4 C3×C42⋊C2 C6×M4(2) Q8×C2×C6 C23.38D4 C6×Q8 Q8⋊C4 C42⋊C2 C2×M4(2) C22×Q8 C2×Q8 C2×C12 C22×C6 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 3 1 6 2 2 4

Matrix representation of C3×C23.38D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 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 1 0 72 0 0 0 1 0 0 72
,
 72 0 0 0 0 0 0 72 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 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 57 67 0 0 0 0 6 16 0 0 0 0 0 0 5 0 46 17 0 0 50 0 51 68 0 0 50 23 23 45 0 0 5 45 23 45
,
 6 16 0 0 0 0 57 67 0 0 0 0 0 0 1 0 71 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 1 72 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[57,6,0,0,0,0,67,16,0,0,0,0,0,0,5,50,50,5,0,0,0,0,23,45,0,0,46,51,23,23,0,0,17,68,45,45],[6,57,0,0,0,0,16,67,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,71,72,72,72,0,0,0,1,0,0] >;

C3×C23.38D4 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{38}D_4
% in TeX

G:=Group("C3xC2^3.38D4");
// GroupNames label

G:=SmallGroup(192,852);
// by ID

G=gap.SmallGroup(192,852);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,1059,520,4204,2111,172]);
// Polycyclic

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

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