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## G = C3×(C22×C8)⋊C2order 192 = 26·3

### Direct product of C3 and (C22×C8)⋊C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×(C22×C8)⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C2×C24 — C3×C22⋊C8 — C3×(C22×C8)⋊C2
 Lower central C1 — C22 — C3×(C22×C8)⋊C2
 Upper central C1 — C2×C12 — C3×(C22×C8)⋊C2

Generators and relations for C3×(C22×C8)⋊C2
G = < a,b,c,d,e | a3=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bd4, ede=cd=dc, ce=ec >

Subgroups: 242 in 158 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C2×C24, C2×C24, C3×M4(2), C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, (C22×C8)⋊C2, C3×C22⋊C8, C22×C24, C6×M4(2), C6×C4○D4, C3×(C22×C8)⋊C2
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○D4, C3×C22⋊C4, C22×C12, C6×D4, (C22×C8)⋊C2, C6×C22⋊C4, C3×C8○D4, C3×(C22×C8)⋊C2

Smallest permutation representation of C3×(C22×C8)⋊C2
On 96 points
Generators in S96
(1 67 19)(2 68 20)(3 69 21)(4 70 22)(5 71 23)(6 72 24)(7 65 17)(8 66 18)(9 30 60)(10 31 61)(11 32 62)(12 25 63)(13 26 64)(14 27 57)(15 28 58)(16 29 59)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 53 89)(42 54 90)(43 55 91)(44 56 92)(45 49 93)(46 50 94)(47 51 95)(48 52 96)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 85)(50 86)(51 87)(52 88)(53 81)(54 82)(55 83)(56 84)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 65)(64 66)
(1 79)(2 80)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 89)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(65 85)(66 86)(67 87)(68 88)(69 81)(70 82)(71 83)(72 84)
(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 25)(2 46)(3 27)(4 48)(5 29)(6 42)(7 31)(8 44)(9 36)(10 17)(11 38)(12 19)(13 40)(14 21)(15 34)(16 23)(18 92)(20 94)(22 96)(24 90)(26 80)(28 74)(30 76)(32 78)(33 95)(35 89)(37 91)(39 93)(41 75)(43 77)(45 79)(47 73)(49 87)(50 68)(51 81)(52 70)(53 83)(54 72)(55 85)(56 66)(57 69)(58 82)(59 71)(60 84)(61 65)(62 86)(63 67)(64 88)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C6 C6 C6 C6 C12 C12 D4 C3×D4 C8○D4 C3×C8○D4 kernel C3×(C22×C8)⋊C2 C3×C22⋊C8 C22×C24 C6×M4(2) C6×C4○D4 (C22×C8)⋊C2 C6×D4 C6×Q8 C22⋊C8 C22×C8 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C2×C12 C2×C4 C6 C2 # reps 1 4 1 1 1 2 6 2 8 2 2 2 12 4 4 8 8 16

Matrix representation of C3×(C22×C8)⋊C2 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 1 0
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 46 0 0 0 71 27 0 0 0 0 10 0 0 0 0 10
,
 1 46 0 0 0 72 0 0 0 0 0 27 0 0 46 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[46,71,0,0,0,27,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,46,72,0,0,0,0,0,46,0,0,27,0] >;

C3×(C22×C8)⋊C2 in GAP, Magma, Sage, TeX

C_3\times (C_2^2\times C_8)\rtimes C_2
% in TeX

G:=Group("C3x(C2^2xC8):C2");
// GroupNames label

G:=SmallGroup(192,841);
// by ID

G=gap.SmallGroup(192,841);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,1059,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*d^4,e*d*e=c*d=d*c,c*e=e*c>;
// generators/relations

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