Copied to
clipboard

## G = (C6×D4).11C4order 192 = 26·3

### 5th non-split extension by C6×D4 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C6×D4).11C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — (C6×D4).11C4
 Lower central C3 — C2×C6 — (C6×D4).11C4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for (C6×D4).11C4
G = < a,b,c,d | a6=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=a3b2c >

Subgroups: 296 in 158 conjugacy classes, 71 normal (27 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, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, 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×C3⋊C8, C2×C3⋊C8, C4.Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, (C22×C8)⋊C2, C12.55D4, C22×C3⋊C8, C2×C4.Dic3, C6×C4○D4, (C6×D4).11C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C8○D4, C6.D4, C22×Dic3, C2×C3⋊D4, (C22×C8)⋊C2, D4.Dic3, C2×C6.D4, (C6×D4).11C4

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C2 C4 C4 S3 D4 D6 Dic3 Dic3 C3⋊D4 C8○D4 D4.Dic3 kernel (C6×D4).11C4 C12.55D4 C22×C3⋊C8 C2×C4.Dic3 C6×C4○D4 C6×D4 C6×Q8 C2×C4○D4 C2×C12 C22×C4 C2×D4 C2×Q8 C2×C4 C6 C2 # reps 1 4 1 1 1 6 2 1 4 3 3 1 8 8 4

Matrix representation of (C6×D4).11C4 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 65 0 0 0 24 9
,
 46 54 0 0 0 27 0 0 0 0 1 0 0 0 0 1
,
 46 54 0 0 46 27 0 0 0 0 72 0 0 0 20 1
,
 51 0 0 0 0 51 0 0 0 0 19 53 0 0 40 54
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,65,24,0,0,0,9],[46,0,0,0,54,27,0,0,0,0,1,0,0,0,0,1],[46,46,0,0,54,27,0,0,0,0,72,20,0,0,0,1],[51,0,0,0,0,51,0,0,0,0,19,40,0,0,53,54] >;

(C6×D4).11C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)._{11}C_4
% in TeX

G:=Group("(C6xD4).11C4");
// GroupNames label

G:=SmallGroup(192,793);
// by ID

G=gap.SmallGroup(192,793);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^3*b^2*c>;
// generators/relations

׿
×
𝔽