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G = C4×D24order 192 = 26·3

Direct product of C4 and D24

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×D24, C125D8, C42.259D6, C31(C4×D8), (C4×C8)⋊7S3, C810(C4×S3), C6.2(C2×D8), C6.7(C4×D4), (C4×C24)⋊12C2, C2422(C2×C4), (C4×D12)⋊1C2, D128(C2×C4), C2.1(C2×D24), C241C428C2, C6.3(C4○D8), C2.10(C4×D12), (C2×C8).286D6, (C2×C4).61D12, (C2×D24).14C2, C2.2(C4○D24), (C2×C12).351D4, C2.D2443C2, C22.28(C2×D12), C12.217(C4○D4), C4.101(C4○D12), C12.102(C22×C4), (C2×C24).346C22, (C2×C12).721C23, (C4×C12).326C22, (C2×D12).188C22, C4⋊Dic3.263C22, C4.60(S3×C2×C4), (C2×C6).104(C2×D4), (C2×C4).664(C22×S3), SmallGroup(192,251)

Series: Derived Chief Lower central Upper central

C1C12 — C4×D24
C1C3C6C2×C6C2×C12C2×D12C2×D24 — C4×D24
C3C6C12 — C4×D24
C1C2×C4C42C4×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 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C8A···8H12A···12L24A···24P
order1222222234444444444446668···812···1224···24
size111112121212211112222121212122222···22···22···2

60 irreducible representations

dim1111111222222222222
type+++++++++++++
imageC1C2C2C2C2C2C4S3D4D6D6D8C4○D4C4×S3D12C4○D8D24C4○D12C4○D24
kernelC4×D24C241C4C2.D24C4×C24C4×D12C2×D24D24C4×C8C2×C12C42C2×C8C12C12C8C2×C4C6C4C4C2
# reps1121218121242444848

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

4600
0460
0046
,
100
06850
02318
,
100
0147
06659
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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