Copied to
clipboard

## G = C2×C4○D24order 192 = 26·3

### Direct product of C2 and C4○D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C4○D24
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×C4○D12 — C2×C4○D24
 Lower central C3 — C6 — C12 — C2×C4○D24
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C2×C4○D24
G = < a,b,c,d | a2=b4=d2=1, c12=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c11 >

Subgroups: 728 in 266 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C24⋊C2, D24, Dic12, C2×C24, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C2×C4○D8, C2×C24⋊C2, C2×D24, C4○D24, C2×Dic12, C22×C24, C2×C4○D12, C2×C4○D24
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C4○D8, C22×D4, C2×D12, S3×C23, C2×C4○D8, C4○D24, C22×D12, C2×C4○D24

Smallest permutation representation of C2×C4○D24
On 96 points
Generators in S96
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 25)(21 26)(22 27)(23 28)(24 29)(49 93)(50 94)(51 95)(52 96)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)
(1 73 13 85)(2 74 14 86)(3 75 15 87)(4 76 16 88)(5 77 17 89)(6 78 18 90)(7 79 19 91)(8 80 20 92)(9 81 21 93)(10 82 22 94)(11 83 23 95)(12 84 24 96)(25 72 37 60)(26 49 38 61)(27 50 39 62)(28 51 40 63)(29 52 41 64)(30 53 42 65)(31 54 43 66)(32 55 44 67)(33 56 45 68)(34 57 46 69)(35 58 47 70)(36 59 48 71)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(49 76)(50 75)(51 74)(52 73)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D12 D12 C4○D8 C4○D24 kernel C2×C4○D24 C2×C24⋊C2 C2×D24 C4○D24 C2×Dic12 C22×C24 C2×C4○D12 C22×C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C4 C23 C6 C2 # reps 1 2 1 8 1 1 2 1 3 1 6 1 6 2 8 16

Matrix representation of C2×C4○D24 in GL3(𝔽73) generated by

 72 0 0 0 1 0 0 0 1
,
 72 0 0 0 46 0 0 0 46
,
 72 0 0 0 50 18 0 55 68
,
 72 0 0 0 55 68 0 50 18
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[72,0,0,0,46,0,0,0,46],[72,0,0,0,50,55,0,18,68],[72,0,0,0,55,50,0,68,18] >;

C2×C4○D24 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_{24}
% in TeX

G:=Group("C2xC4oD24");
// GroupNames label

G:=SmallGroup(192,1300);
// by ID

G=gap.SmallGroup(192,1300);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,80,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^12=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^11>;
// generators/relations

׿
×
𝔽