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G = C4×D24order 192 = 26·3

Direct product of C4 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4×D24
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×D24 — C4×D24
 Lower central C3 — C6 — C12 — C4×D24
 Upper central C1 — C2×C4 — C42 — C4×C8

Generators and relations for C4×D24
G = < a,b,c | a4=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 456 in 134 conjugacy classes, 55 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], S3 [×4], C6 [×3], C8 [×2], C8, C2×C4 [×3], C2×C4 [×6], D4 [×6], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], C12, D6 [×8], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], C24 [×2], C24, C4×S3 [×4], D12 [×4], D12 [×2], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, D24 [×4], C4⋊Dic3 [×2], D6⋊C4 [×2], C4×C12, C2×C24 [×2], S3×C2×C4 [×2], C2×D12 [×2], C4×D8, C241C4, C2.D24 [×2], C4×C24, C4×D12 [×2], C2×D24, C4×D24
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], D8 [×2], C22×C4, C2×D4, C4○D4, C4×S3 [×2], D12 [×2], C22×S3, C4×D4, C2×D8, C4○D8, D24 [×2], S3×C2×C4, C2×D12, C4○D12, C4×D8, C4×D12, C2×D24, C4○D24, C4×D24

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 8A ··· 8H 12A ··· 12L 24A ··· 24P order 1 2 2 2 2 2 2 2 3 4 4 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 12 12 2 1 1 1 1 2 2 2 2 12 12 12 12 2 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 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D8 C4○D4 C4×S3 D12 C4○D8 D24 C4○D12 C4○D24 kernel C4×D24 C24⋊1C4 C2.D24 C4×C24 C4×D12 C2×D24 D24 C4×C8 C2×C12 C42 C2×C8 C12 C12 C8 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 8 1 2 1 2 4 2 4 4 4 8 4 8

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

 46 0 0 0 46 0 0 0 46
,
 1 0 0 0 68 50 0 23 18
,
 1 0 0 0 14 7 0 66 59
G:=sub<GL(3,GF(73))| [46,0,0,0,46,0,0,0,46],[1,0,0,0,68,23,0,50,18],[1,0,0,0,14,66,0,7,59] >;

C4×D24 in GAP, Magma, Sage, TeX

C_4\times D_{24}
% in TeX

G:=Group("C4xD24");
// GroupNames label

G:=SmallGroup(192,251);
// by ID

G=gap.SmallGroup(192,251);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,58,1684,102,6278]);
// 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^-1>;
// generators/relations

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