Copied to
clipboard

## G = (C6×C12).C4order 288 = 25·32

### 13rd non-split extension by C6×C12 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — (C6×C12).C4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C12.58D6 — (C6×C12).C4
 Lower central C32 — C3×C6 — C62 — (C6×C12).C4
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for (C6×C12).C4
G = < a,b,c | a6=b12=1, c4=b6, ab=ba, cac-1=a-1b6, cbc-1=a3b-1 >

Subgroups: 268 in 114 conjugacy classes, 57 normal (11 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, C12 [×8], C12 [×8], C2×C6 [×4], M4(2) [×2], C2×Q8, C3×C6, C3×C6, C3⋊C8 [×8], C2×C12 [×12], C3×Q8 [×8], C4.10D4, C3×C12 [×2], C3×C12 [×2], C62, C4.Dic3 [×8], C6×Q8 [×4], C324C8 [×2], C6×C12, C6×C12 [×2], Q8×C32 [×2], C12.10D4 [×4], C12.58D6 [×2], Q8×C3×C6, (C6×C12).C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], Dic3 [×8], D6 [×4], C22⋊C4, C3⋊S3, C2×Dic3 [×4], C3⋊D4 [×8], C4.10D4, C3⋊Dic3 [×2], C2×C3⋊S3, C6.D4 [×4], C2×C3⋊Dic3, C327D4 [×2], C12.10D4 [×4], C625C4, (C6×C12).C4

Smallest permutation representation of (C6×C12).C4
On 144 points
Generators in S144
(1 124 43)(2 125 44)(3 126 45)(4 127 46)(5 128 47)(6 129 48)(7 130 37)(8 131 38)(9 132 39)(10 121 40)(11 122 41)(12 123 42)(13 94 59)(14 95 60)(15 96 49)(16 85 50)(17 86 51)(18 87 52)(19 88 53)(20 89 54)(21 90 55)(22 91 56)(23 92 57)(24 93 58)(25 119 76 31 113 82)(26 120 77 32 114 83)(27 109 78 33 115 84)(28 110 79 34 116 73)(29 111 80 35 117 74)(30 112 81 36 118 75)(61 133 105 67 139 99)(62 134 106 68 140 100)(63 135 107 69 141 101)(64 136 108 70 142 102)(65 137 97 71 143 103)(66 138 98 72 144 104)
(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)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 118 94 140 7 112 88 134)(2 117 89 133 8 111 95 139)(3 116 96 138 9 110 90 144)(4 115 91 143 10 109 85 137)(5 114 86 136 11 120 92 142)(6 113 93 141 12 119 87 135)(13 106 37 75 19 100 43 81)(14 105 44 80 20 99 38 74)(15 104 39 73 21 98 45 79)(16 103 46 78 22 97 40 84)(17 102 41 83 23 108 47 77)(18 101 48 76 24 107 42 82)(25 58 63 123 31 52 69 129)(26 51 70 122 32 57 64 128)(27 56 65 121 33 50 71 127)(28 49 72 132 34 55 66 126)(29 54 67 131 35 60 61 125)(30 59 62 130 36 53 68 124)

G:=sub<Sym(144)| (1,124,43)(2,125,44)(3,126,45)(4,127,46)(5,128,47)(6,129,48)(7,130,37)(8,131,38)(9,132,39)(10,121,40)(11,122,41)(12,123,42)(13,94,59)(14,95,60)(15,96,49)(16,85,50)(17,86,51)(18,87,52)(19,88,53)(20,89,54)(21,90,55)(22,91,56)(23,92,57)(24,93,58)(25,119,76,31,113,82)(26,120,77,32,114,83)(27,109,78,33,115,84)(28,110,79,34,116,73)(29,111,80,35,117,74)(30,112,81,36,118,75)(61,133,105,67,139,99)(62,134,106,68,140,100)(63,135,107,69,141,101)(64,136,108,70,142,102)(65,137,97,71,143,103)(66,138,98,72,144,104), (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)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,118,94,140,7,112,88,134)(2,117,89,133,8,111,95,139)(3,116,96,138,9,110,90,144)(4,115,91,143,10,109,85,137)(5,114,86,136,11,120,92,142)(6,113,93,141,12,119,87,135)(13,106,37,75,19,100,43,81)(14,105,44,80,20,99,38,74)(15,104,39,73,21,98,45,79)(16,103,46,78,22,97,40,84)(17,102,41,83,23,108,47,77)(18,101,48,76,24,107,42,82)(25,58,63,123,31,52,69,129)(26,51,70,122,32,57,64,128)(27,56,65,121,33,50,71,127)(28,49,72,132,34,55,66,126)(29,54,67,131,35,60,61,125)(30,59,62,130,36,53,68,124)>;

G:=Group( (1,124,43)(2,125,44)(3,126,45)(4,127,46)(5,128,47)(6,129,48)(7,130,37)(8,131,38)(9,132,39)(10,121,40)(11,122,41)(12,123,42)(13,94,59)(14,95,60)(15,96,49)(16,85,50)(17,86,51)(18,87,52)(19,88,53)(20,89,54)(21,90,55)(22,91,56)(23,92,57)(24,93,58)(25,119,76,31,113,82)(26,120,77,32,114,83)(27,109,78,33,115,84)(28,110,79,34,116,73)(29,111,80,35,117,74)(30,112,81,36,118,75)(61,133,105,67,139,99)(62,134,106,68,140,100)(63,135,107,69,141,101)(64,136,108,70,142,102)(65,137,97,71,143,103)(66,138,98,72,144,104), (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)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,118,94,140,7,112,88,134)(2,117,89,133,8,111,95,139)(3,116,96,138,9,110,90,144)(4,115,91,143,10,109,85,137)(5,114,86,136,11,120,92,142)(6,113,93,141,12,119,87,135)(13,106,37,75,19,100,43,81)(14,105,44,80,20,99,38,74)(15,104,39,73,21,98,45,79)(16,103,46,78,22,97,40,84)(17,102,41,83,23,108,47,77)(18,101,48,76,24,107,42,82)(25,58,63,123,31,52,69,129)(26,51,70,122,32,57,64,128)(27,56,65,121,33,50,71,127)(28,49,72,132,34,55,66,126)(29,54,67,131,35,60,61,125)(30,59,62,130,36,53,68,124) );

G=PermutationGroup([(1,124,43),(2,125,44),(3,126,45),(4,127,46),(5,128,47),(6,129,48),(7,130,37),(8,131,38),(9,132,39),(10,121,40),(11,122,41),(12,123,42),(13,94,59),(14,95,60),(15,96,49),(16,85,50),(17,86,51),(18,87,52),(19,88,53),(20,89,54),(21,90,55),(22,91,56),(23,92,57),(24,93,58),(25,119,76,31,113,82),(26,120,77,32,114,83),(27,109,78,33,115,84),(28,110,79,34,116,73),(29,111,80,35,117,74),(30,112,81,36,118,75),(61,133,105,67,139,99),(62,134,106,68,140,100),(63,135,107,69,141,101),(64,136,108,70,142,102),(65,137,97,71,143,103),(66,138,98,72,144,104)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,118,94,140,7,112,88,134),(2,117,89,133,8,111,95,139),(3,116,96,138,9,110,90,144),(4,115,91,143,10,109,85,137),(5,114,86,136,11,120,92,142),(6,113,93,141,12,119,87,135),(13,106,37,75,19,100,43,81),(14,105,44,80,20,99,38,74),(15,104,39,73,21,98,45,79),(16,103,46,78,22,97,40,84),(17,102,41,83,23,108,47,77),(18,101,48,76,24,107,42,82),(25,58,63,123,31,52,69,129),(26,51,70,122,32,57,64,128),(27,56,65,121,33,50,71,127),(28,49,72,132,34,55,66,126),(29,54,67,131,35,60,61,125),(30,59,62,130,36,53,68,124)])

51 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A 8B 8C 8D 12A ··· 12X order 1 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 2 2 2 2 2 2 2 4 4 2 ··· 2 36 36 36 36 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + - image C1 C2 C2 C4 S3 D4 Dic3 D6 C3⋊D4 C4.10D4 C12.10D4 kernel (C6×C12).C4 C12.58D6 Q8×C3×C6 C6×C12 C6×Q8 C3×C12 C2×C12 C2×C12 C12 C32 C3 # reps 1 2 1 4 4 2 8 4 16 1 8

Matrix representation of (C6×C12).C4 in GL6(𝔽73)

 65 0 0 0 0 0 0 9 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 64 0 0 0 0 0 0 65 0 0 0 0 0 0 56 31 0 0 0 0 31 17 0 0 0 0 0 0 10 29 0 0 0 0 29 63
,
 0 47 0 0 0 0 14 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 72 0 0 0

G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,9,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[64,0,0,0,0,0,0,65,0,0,0,0,0,0,56,31,0,0,0,0,31,17,0,0,0,0,0,0,10,29,0,0,0,0,29,63],[0,14,0,0,0,0,47,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C6×C12).C4 in GAP, Magma, Sage, TeX

(C_6\times C_{12}).C_4
% in TeX

G:=Group("(C6xC12).C4");
// GroupNames label

G:=SmallGroup(288,311);
// by ID

G=gap.SmallGroup(288,311);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,120,219,100,675,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^12=1,c^4=b^6,a*b=b*a,c*a*c^-1=a^-1*b^6,c*b*c^-1=a^3*b^-1>;
// generators/relations

׿
×
𝔽