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G = C6×D4⋊C4order 192 = 26·3

Direct product of C6 and D4⋊C4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×D4⋊C4
G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 402 in 202 conjugacy classes, 98 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, D4, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C22×C12, C22×C12, C6×D4, C6×D4, C23×C6, C2×D4⋊C4, C3×D4⋊C4, C6×C4⋊C4, C22×C24, D4×C2×C6, C6×D4⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, D8, SD16, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C3×C22⋊C4, C3×D8, C3×SD16, C22×C12, C6×D4, C2×D4⋊C4, C3×D4⋊C4, C6×C22⋊C4, C6×D8, C6×SD16, C6×D4⋊C4

Smallest permutation representation of C6×D4⋊C4
On 96 points
Generators in S96
(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 22 62 89)(2 23 63 90)(3 24 64 85)(4 19 65 86)(5 20 66 87)(6 21 61 88)(7 50 43 30)(8 51 44 25)(9 52 45 26)(10 53 46 27)(11 54 47 28)(12 49 48 29)(13 34 91 72)(14 35 92 67)(15 36 93 68)(16 31 94 69)(17 32 95 70)(18 33 96 71)(37 78 82 58)(38 73 83 59)(39 74 84 60)(40 75 79 55)(41 76 80 56)(42 77 81 57)
(1 77)(2 78)(3 73)(4 74)(5 75)(6 76)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(13 52)(14 53)(15 54)(16 49)(17 50)(18 51)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 96)(26 91)(27 92)(28 93)(29 94)(30 95)(43 70)(44 71)(45 72)(46 67)(47 68)(48 69)(55 66)(56 61)(57 62)(58 63)(59 64)(60 65)(79 87)(80 88)(81 89)(82 90)(83 85)(84 86)
(1 91 81 52)(2 92 82 53)(3 93 83 54)(4 94 84 49)(5 95 79 50)(6 96 80 51)(7 20 32 55)(8 21 33 56)(9 22 34 57)(10 23 35 58)(11 24 36 59)(12 19 31 60)(13 42 26 62)(14 37 27 63)(15 38 28 64)(16 39 29 65)(17 40 30 66)(18 41 25 61)(43 87 70 75)(44 88 71 76)(45 89 72 77)(46 90 67 78)(47 85 68 73)(48 86 69 74)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D4 D4 D8 SD16 C3×D4 C3×D4 C3×D8 C3×SD16 kernel C6×D4⋊C4 C3×D4⋊C4 C6×C4⋊C4 C22×C24 D4×C2×C6 C2×D4⋊C4 C6×D4 D4⋊C4 C2×C4⋊C4 C22×C8 C22×D4 C2×D4 C2×C12 C22×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 2 8 8 2 2 2 16 3 1 4 4 6 2 8 8

Matrix representation of C6×D4⋊C4 in GL4(𝔽73) generated by

 64 0 0 0 0 72 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 0 72 0 0 0 0 0 1 0 0 1 0
,
 27 0 0 0 0 1 0 0 0 0 16 57 0 0 57 57
G:=sub<GL(4,GF(73))| [64,0,0,0,0,72,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[27,0,0,0,0,1,0,0,0,0,16,57,0,0,57,57] >;

C6×D4⋊C4 in GAP, Magma, Sage, TeX

C_6\times D_4\rtimes C_4
% in TeX

G:=Group("C6xD4:C4");
// GroupNames label

G:=SmallGroup(192,847);
// by ID

G=gap.SmallGroup(192,847);
# by ID

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

G:=Group<a,b,c,d|a^6=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations

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