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G = D4xC24order 192 = 26·3

Direct product of C24 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4xC24, (C4xC8):4C6, C4:C8:18C6, C12:6(C2xC8), (C4xC24):9C2, C4:1(C2xC24), (C22xC8):9C6, C4.79(C6xD4), C2.3(D4xC12), C4:C4.11C12, C22:C8:15C6, C22:2(C2xC24), (C22xC24):9C2, (C4xD4).14C6, (C6xD4).23C4, C6.111(C4xD4), (C2xD4).11C12, (D4xC12).29C2, C6.48(C8oD4), C12.484(C2xD4), C22:C4.7C12, C42.68(C2xC6), C6.33(C22xC8), C2.4(C22xC24), C23.23(C2xC12), C12.353(C4oD4), (C2xC12).990C23, (C4xC12).353C22, (C2xC24).361C22, C22.22(C22xC12), (C22xC12).499C22, (C2xC6):4(C2xC8), (C3xC4:C8):37C2, C2.2(C3xC8oD4), (C3xC4:C4).23C4, C4.51(C3xC4oD4), (C3xC22:C8):32C2, (C2xC8).107(C2xC6), (C2xC4).36(C2xC12), (C2xC12).212(C2xC4), (C3xC22:C4).14C4, (C22xC6).85(C2xC4), (C2xC4).158(C22xC6), (C2xC6).240(C22xC4), (C22xC4).102(C2xC6), SmallGroup(192,867)

Series: Derived Chief Lower central Upper central

C1C2 — D4xC24
C1C2C4C2xC4C2xC12C2xC24C3xC22:C8 — D4xC24
C1C2 — D4xC24
C1C2xC24 — D4xC24

Generators and relations for D4xC24
 G = < a,b,c | a24=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 178 in 134 conjugacy classes, 90 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2xC4, C2xC4, C2xC4, D4, C23, C12, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xC8, C2xC8, C22xC4, C2xD4, C24, C24, C2xC12, C2xC12, C2xC12, C3xD4, C22xC6, C4xC8, C22:C8, C4:C8, C4xD4, C22xC8, C4xC12, C3xC22:C4, C3xC4:C4, C2xC24, C2xC24, C2xC24, C22xC12, C6xD4, C8xD4, C4xC24, C3xC22:C8, C3xC4:C8, D4xC12, C22xC24, D4xC24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2xC4, D4, C23, C12, C2xC6, C2xC8, C22xC4, C2xD4, C4oD4, C24, C2xC12, C3xD4, C22xC6, C4xD4, C22xC8, C8oD4, C2xC24, C22xC12, C6xD4, C3xC4oD4, C8xD4, D4xC12, C22xC24, C3xC8oD4, D4xC24

Smallest permutation representation of D4xC24
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 61 80 34)(2 62 81 35)(3 63 82 36)(4 64 83 37)(5 65 84 38)(6 66 85 39)(7 67 86 40)(8 68 87 41)(9 69 88 42)(10 70 89 43)(11 71 90 44)(12 72 91 45)(13 49 92 46)(14 50 93 47)(15 51 94 48)(16 52 95 25)(17 53 96 26)(18 54 73 27)(19 55 74 28)(20 56 75 29)(21 57 76 30)(22 58 77 31)(23 59 78 32)(24 60 79 33)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(49 92)(50 93)(51 94)(52 95)(53 96)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E···4L6A···6F6G···6N8A···8H8I···8T12A···12H12I···12X24A···24P24Q···24AN
order122222223344444···46···66···68···88···812···1212···1224···2424···24
size111122221111112···21···12···21···12···21···12···21···12···2

120 irreducible representations

dim11111111111111111111222222
type+++++++
imageC1C2C2C2C2C2C3C4C4C4C6C6C6C6C6C8C12C12C12C24D4C4oD4C3xD4C8oD4C3xC4oD4C3xC8oD4
kernelD4xC24C4xC24C3xC22:C8C3xC4:C8D4xC12C22xC24C8xD4C3xC22:C4C3xC4:C4C6xD4C4xC8C22:C8C4:C8C4xD4C22xC8C3xD4C22:C4C4:C4C2xD4D4C24C12C8C6C4C2
# reps1121122422242241684432224448

Matrix representation of D4xC24 in GL3(F73) generated by

2200
0700
0070
,
7200
0119
04672
,
100
0720
0271
G:=sub<GL(3,GF(73))| [22,0,0,0,70,0,0,0,70],[72,0,0,0,1,46,0,19,72],[1,0,0,0,72,27,0,0,1] >;

D4xC24 in GAP, Magma, Sage, TeX

D_4\times C_{24}
% in TeX

G:=Group("D4xC24");
// GroupNames label

G:=SmallGroup(192,867);
// by ID

G=gap.SmallGroup(192,867);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,268,124]);
// Polycyclic

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

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