Copied to
clipboard

## G = C4×C24⋊C2order 192 = 26·3

### Direct product of C4 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4×C24⋊C2
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×C24⋊C2 — C4×C24⋊C2
 Lower central C3 — C6 — C12 — C4×C24⋊C2
 Upper central C1 — C2×C4 — C42 — C4×C8

Generators and relations for C4×C24⋊C2
G = < a,b,c | a4=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 360 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C24⋊C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4×SD16, C2.Dic12, C8⋊Dic3, C2.D24, C4×C24, C4×Dic6, C4×D12, C2×C24⋊C2, C4×C24⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, SD16, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C2×SD16, C4○D8, C24⋊C2, S3×C2×C4, C2×D12, C4○D12, C4×SD16, C4×D12, C2×C24⋊C2, C4○D24, C4×C24⋊C2

Smallest permutation representation of C4×C24⋊C2
On 96 points
Generators in S96
(1 30 51 86)(2 31 52 87)(3 32 53 88)(4 33 54 89)(5 34 55 90)(6 35 56 91)(7 36 57 92)(8 37 58 93)(9 38 59 94)(10 39 60 95)(11 40 61 96)(12 41 62 73)(13 42 63 74)(14 43 64 75)(15 44 65 76)(16 45 66 77)(17 46 67 78)(18 47 68 79)(19 48 69 80)(20 25 70 81)(21 26 71 82)(22 27 72 83)(23 28 49 84)(24 29 50 85)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 47)(26 34)(27 45)(28 32)(29 43)(31 41)(33 39)(35 37)(36 48)(38 46)(40 44)(49 53)(50 64)(52 62)(54 60)(55 71)(56 58)(57 69)(59 67)(61 65)(66 72)(68 70)(73 87)(75 85)(76 96)(77 83)(78 94)(79 81)(80 92)(82 90)(84 88)(89 95)(91 93)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 SD16 C4○D4 C4×S3 D12 C4○D8 C24⋊C2 C4○D12 C4○D24 kernel C4×C24⋊C2 C2.Dic12 C8⋊Dic3 C2.D24 C4×C24 C4×Dic6 C4×D12 C2×C24⋊C2 C24⋊C2 C4×C8 C2×C12 C42 C2×C8 C12 C12 C8 C2×C4 C6 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 2 4 2 4 4 4 8 4 8

Matrix representation of C4×C24⋊C2 in GL3(𝔽73) generated by

 46 0 0 0 27 0 0 0 27
,
 1 0 0 0 62 37 0 36 25
,
 72 0 0 0 1 0 0 72 72
G:=sub<GL(3,GF(73))| [46,0,0,0,27,0,0,0,27],[1,0,0,0,62,36,0,37,25],[72,0,0,0,1,72,0,0,72] >;

C4×C24⋊C2 in GAP, Magma, Sage, TeX

C_4\times C_{24}\rtimes C_2
% in TeX

G:=Group("C4xC24:C2");
// GroupNames label

G:=SmallGroup(192,250);
// by ID

G=gap.SmallGroup(192,250);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,58,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^4=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations

׿
×
𝔽