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## G = C3×D8⋊C4order 192 = 26·3

### Direct product of C3 and D8⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×D8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×D4⋊C4 — C3×D8⋊C4
 Lower central C1 — C2 — C4 — C3×D8⋊C4
 Upper central C1 — C2×C6 — C4×C12 — C3×D8⋊C4

Generators and relations for C3×D8⋊C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >

Subgroups: 250 in 132 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×D8, C22×C12, C6×D4, D8⋊C4, C3×C8⋊C4, C3×D4⋊C4, C3×C4.Q8, D4×C12, C6×D8, C3×D8⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C8⋊C22, C22×C12, C6×D4, C3×C4○D4, D8⋊C4, D4×C12, C3×C8⋊C22, C3×D8⋊C4

Smallest permutation representation of C3×D8⋊C4
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 68 84)(10 69 85)(11 70 86)(12 71 87)(13 72 88)(14 65 81)(15 66 82)(16 67 83)(17 28 55)(18 29 56)(19 30 49)(20 31 50)(21 32 51)(22 25 52)(23 26 53)(24 27 54)(33 42 57)(34 43 58)(35 44 59)(36 45 60)(37 46 61)(38 47 62)(39 48 63)(40 41 64)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 22)(18 21)(19 20)(23 24)(25 28)(26 27)(29 32)(30 31)(33 34)(35 40)(36 39)(37 38)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)
(1 71 27 43)(2 68 28 48)(3 65 29 45)(4 70 30 42)(5 67 31 47)(6 72 32 44)(7 69 25 41)(8 66 26 46)(9 17 39 96)(10 22 40 93)(11 19 33 90)(12 24 34 95)(13 21 35 92)(14 18 36 89)(15 23 37 94)(16 20 38 91)(49 57 74 86)(50 62 75 83)(51 59 76 88)(52 64 77 85)(53 61 78 82)(54 58 79 87)(55 63 80 84)(56 60 73 81)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6N 8A 8B 8C 8D 12A ··· 12L 12M ··· 12T 24A ··· 24H order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 4 4 1 1 2 ··· 2 4 4 4 4 1 ··· 1 4 ··· 4 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 1 1 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 D4 C4○D4 C3×D4 C3×C4○D4 C8⋊C22 C3×C8⋊C22 kernel C3×D8⋊C4 C3×C8⋊C4 C3×D4⋊C4 C3×C4.Q8 D4×C12 C6×D8 D8⋊C4 C3×D8 C8⋊C4 D4⋊C4 C4.Q8 C4×D4 C2×D8 D8 C2×C12 C12 C2×C4 C4 C6 C2 # reps 1 1 2 1 2 1 2 8 2 4 2 4 2 16 2 2 4 4 2 4

Matrix representation of C3×D8⋊C4 in GL8(𝔽73)

 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 2 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 17 46 0 0 0 0 0 0 54 56 0 0 0 0 0 0 0 0 0 33 0 24 0 0 0 0 20 33 61 24 0 0 0 0 0 49 0 40 0 0 0 0 12 49 53 40
,
 1 2 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 56 54 0 0 0 0 0 0 19 17 0 0 0 0 0 0 0 0 0 33 0 24 0 0 0 0 53 0 12 0 0 0 0 0 0 49 0 40 0 0 0 0 61 0 20 0
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 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 1 0 0

G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,0,2,72,0,0,0,0,0,0,0,0,17,54,0,0,0,0,0,0,46,56,0,0,0,0,0,0,0,0,0,20,0,12,0,0,0,0,33,33,49,49,0,0,0,0,0,61,0,53,0,0,0,0,24,24,40,40],[1,0,0,0,0,0,0,0,2,72,0,0,0,0,0,0,0,0,56,19,0,0,0,0,0,0,54,17,0,0,0,0,0,0,0,0,0,53,0,61,0,0,0,0,33,0,49,0,0,0,0,0,0,12,0,20,0,0,0,0,24,0,40,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C3×D8⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes C_4
% in TeX

G:=Group("C3xD8:C4");
// GroupNames label

G:=SmallGroup(192,875);
// by ID

G=gap.SmallGroup(192,875);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,772,4204,2111,172]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations

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