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G = C3×Q8⋊5D4order 192 = 26·3

Direct product of C3 and Q8⋊5D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×Q8⋊5D4
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊D4 — C3×Q8⋊5D4
 Lower central C1 — C22 — C3×Q8⋊5D4
 Upper central C1 — C2×C6 — C3×Q8⋊5D4

Generators and relations for C3×Q85D4
G = < a,b,c,d,e | a3=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 426 in 290 conjugacy classes, 166 normal (28 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C6×Q8, C6×Q8, C3×C4○D4, Q85D4, D4×C12, Q8×C12, C3×C4⋊D4, C3×C22⋊Q8, C3×C4.4D4, Q8×C2×C6, C6×C4○D4, C3×Q85D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C24, C3×D4, C22×C6, C22×D4, C2×C4○D4, 2- 1+4, C6×D4, C3×C4○D4, C23×C6, Q85D4, D4×C2×C6, C6×C4○D4, C3×2- 1+4, C3×Q85D4

Smallest permutation representation of C3×Q85D4
On 96 points
Generators in S96
(1 26 23)(2 27 24)(3 28 21)(4 25 22)(5 9 18)(6 10 19)(7 11 20)(8 12 17)(13 93 82)(14 94 83)(15 95 84)(16 96 81)(29 38 42)(30 39 43)(31 40 44)(32 37 41)(33 51 46)(34 52 47)(35 49 48)(36 50 45)(53 64 68)(54 61 65)(55 62 66)(56 63 67)(57 73 72)(58 74 69)(59 75 70)(60 76 71)(77 88 92)(78 85 89)(79 86 90)(80 87 91)
(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 64 3 62)(2 63 4 61)(5 57 7 59)(6 60 8 58)(9 73 11 75)(10 76 12 74)(13 42 15 44)(14 41 16 43)(17 69 19 71)(18 72 20 70)(21 55 23 53)(22 54 24 56)(25 65 27 67)(26 68 28 66)(29 95 31 93)(30 94 32 96)(33 85 35 87)(34 88 36 86)(37 81 39 83)(38 84 40 82)(45 79 47 77)(46 78 48 80)(49 91 51 89)(50 90 52 92)
(1 82 6 86)(2 83 7 87)(3 84 8 88)(4 81 5 85)(9 89 25 16)(10 90 26 13)(11 91 27 14)(12 92 28 15)(17 77 21 95)(18 78 22 96)(19 79 23 93)(20 80 24 94)(29 69 47 55)(30 70 48 56)(31 71 45 53)(32 72 46 54)(33 61 37 57)(34 62 38 58)(35 63 39 59)(36 64 40 60)(41 73 51 65)(42 74 52 66)(43 75 49 67)(44 76 50 68)
(1 58)(2 59)(3 60)(4 57)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 42)(14 43)(15 44)(16 41)(17 53)(18 54)(19 55)(20 56)(21 71)(22 72)(23 69)(24 70)(25 73)(26 74)(27 75)(28 76)(29 93)(30 94)(31 95)(32 96)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(45 77)(46 78)(47 79)(48 80)(49 91)(50 92)(51 89)(52 90)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3A 3B 4A ··· 4J 4K ··· 4P 6A ··· 6F 6G 6H 6I 6J 6K ··· 6P 12A ··· 12T 12U ··· 12AF order 1 2 2 2 2 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 6 6 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 4 4 4 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 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 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 2- 1+4 C3×2- 1+4 kernel C3×Q8⋊5D4 D4×C12 Q8×C12 C3×C4⋊D4 C3×C22⋊Q8 C3×C4.4D4 Q8×C2×C6 C6×C4○D4 Q8⋊5D4 C4×D4 C4×Q8 C4⋊D4 C22⋊Q8 C4.4D4 C22×Q8 C2×C4○D4 C3×Q8 C2×C6 Q8 C22 C6 C2 # reps 1 3 1 3 3 3 1 1 2 6 2 6 6 6 2 2 4 4 8 8 1 2

Matrix representation of C3×Q85D4 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 0 1 0 0 12 0 0 0 0 0 1 0 0 0 0 1
,
 0 8 0 0 8 0 0 0 0 0 1 0 0 0 0 1
,
 0 5 0 0 8 0 0 0 0 0 0 1 0 0 12 0
,
 0 8 0 0 5 0 0 0 0 0 12 0 0 0 0 1
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,8,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[0,8,0,0,5,0,0,0,0,0,0,12,0,0,1,0],[0,5,0,0,8,0,0,0,0,0,12,0,0,0,0,1] >;

C3×Q85D4 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_5D_4
% in TeX

G:=Group("C3xQ8:5D4");
// GroupNames label

G:=SmallGroup(192,1437);
// by ID

G=gap.SmallGroup(192,1437);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,344,2102,794,192]);
// Polycyclic

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

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