Copied to
clipboard

## G = C12.38SD16order 192 = 26·3

### 4th non-split extension by C12 of SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.38SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C12⋊2Q8 — C12.38SD16
 Lower central C3 — C6 — C2×C12 — C12.38SD16
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for C12.38SD16
G = < a,b,c | a12=b8=c2=1, bab-1=a-1, ac=ca, cbc=a6b3 >

Subgroups: 280 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×C6, D4⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, D42Q8, C12⋊C8, C12.Q8 [×2], D4⋊Dic3 [×2], C122Q8, D4×C12, C12.38SD16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×SD16, C8⋊C22, D4.S3 [×2], C2×Dic6, C4○D12, C2×C3⋊D4, D42Q8, C12.48D4, C2×D4.S3, D4⋊D6, C12.38SD16

Smallest permutation representation of C12.38SD16
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 82 57 85 47 28 24 71)(2 81 58 96 48 27 13 70)(3 80 59 95 37 26 14 69)(4 79 60 94 38 25 15 68)(5 78 49 93 39 36 16 67)(6 77 50 92 40 35 17 66)(7 76 51 91 41 34 18 65)(8 75 52 90 42 33 19 64)(9 74 53 89 43 32 20 63)(10 73 54 88 44 31 21 62)(11 84 55 87 45 30 22 61)(12 83 56 86 46 29 23 72)
(13 58)(14 59)(15 60)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 1 1 4 4 2 2 2 2 2 4 4 4 24 24 2 2 2 4 4 4 4 12 12 12 12 2 2 2 2 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + - + - + image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 SD16 C4○D4 C3⋊D4 Dic6 C4○D12 C8⋊C22 D4.S3 D4⋊D6 kernel C12.38SD16 C12⋊C8 C12.Q8 D4⋊Dic3 C12⋊2Q8 D4×C12 C4×D4 C2×C12 C3×D4 C42 C4⋊C4 C2×D4 C12 C12 C2×C4 D4 C4 C6 C4 C2 # reps 1 1 2 2 1 1 1 2 2 1 1 1 4 2 4 4 4 1 2 2

Matrix representation of C12.38SD16 in GL4(𝔽73) generated by

 3 0 0 0 0 49 0 0 0 0 72 0 0 0 0 72
,
 0 1 0 0 72 0 0 0 0 0 0 40 0 0 42 12
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 66 72
G:=sub<GL(4,GF(73))| [3,0,0,0,0,49,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,1,0,0,0,0,0,0,42,0,0,40,12],[1,0,0,0,0,1,0,0,0,0,1,66,0,0,0,72] >;

C12.38SD16 in GAP, Magma, Sage, TeX

C_{12}._{38}{\rm SD}_{16}
% in TeX

G:=Group("C12.38SD16");
// GroupNames label

G:=SmallGroup(192,567);
// by ID

G=gap.SmallGroup(192,567);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,253,120,254,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=a^6*b^3>;
// generators/relations

׿
×
𝔽