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## G = C6×C24order 144 = 24·32

### Abelian group of type [6,24]

Aliases: C6×C24, SmallGroup(144,104)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C24
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C3×C24 — C6×C24
 Lower central C1 — C6×C24
 Upper central C1 — C6×C24

Generators and relations for C6×C24
G = < a,b | a6=b24=1, ab=ba >

Subgroups: 66, all normal (14 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, C3×C6, C3×C6 [×2], C24 [×8], C2×C12 [×4], C3×C12 [×2], C62, C2×C24 [×4], C3×C24 [×2], C6×C12, C6×C24
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, C3×C6 [×3], C24 [×8], C2×C12 [×4], C3×C12 [×2], C62, C2×C24 [×4], C3×C24 [×2], C6×C12, C6×C24

Smallest permutation representation of C6×C24
Regular action on 144 points
Generators in S144
(1 49 105 38 125 95)(2 50 106 39 126 96)(3 51 107 40 127 73)(4 52 108 41 128 74)(5 53 109 42 129 75)(6 54 110 43 130 76)(7 55 111 44 131 77)(8 56 112 45 132 78)(9 57 113 46 133 79)(10 58 114 47 134 80)(11 59 115 48 135 81)(12 60 116 25 136 82)(13 61 117 26 137 83)(14 62 118 27 138 84)(15 63 119 28 139 85)(16 64 120 29 140 86)(17 65 97 30 141 87)(18 66 98 31 142 88)(19 67 99 32 143 89)(20 68 100 33 144 90)(21 69 101 34 121 91)(22 70 102 35 122 92)(23 71 103 36 123 93)(24 72 104 37 124 94)
(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)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)

G:=sub<Sym(144)| (1,49,105,38,125,95)(2,50,106,39,126,96)(3,51,107,40,127,73)(4,52,108,41,128,74)(5,53,109,42,129,75)(6,54,110,43,130,76)(7,55,111,44,131,77)(8,56,112,45,132,78)(9,57,113,46,133,79)(10,58,114,47,134,80)(11,59,115,48,135,81)(12,60,116,25,136,82)(13,61,117,26,137,83)(14,62,118,27,138,84)(15,63,119,28,139,85)(16,64,120,29,140,86)(17,65,97,30,141,87)(18,66,98,31,142,88)(19,67,99,32,143,89)(20,68,100,33,144,90)(21,69,101,34,121,91)(22,70,102,35,122,92)(23,71,103,36,123,93)(24,72,104,37,124,94), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)>;

G:=Group( (1,49,105,38,125,95)(2,50,106,39,126,96)(3,51,107,40,127,73)(4,52,108,41,128,74)(5,53,109,42,129,75)(6,54,110,43,130,76)(7,55,111,44,131,77)(8,56,112,45,132,78)(9,57,113,46,133,79)(10,58,114,47,134,80)(11,59,115,48,135,81)(12,60,116,25,136,82)(13,61,117,26,137,83)(14,62,118,27,138,84)(15,63,119,28,139,85)(16,64,120,29,140,86)(17,65,97,30,141,87)(18,66,98,31,142,88)(19,67,99,32,143,89)(20,68,100,33,144,90)(21,69,101,34,121,91)(22,70,102,35,122,92)(23,71,103,36,123,93)(24,72,104,37,124,94), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144) );

G=PermutationGroup([(1,49,105,38,125,95),(2,50,106,39,126,96),(3,51,107,40,127,73),(4,52,108,41,128,74),(5,53,109,42,129,75),(6,54,110,43,130,76),(7,55,111,44,131,77),(8,56,112,45,132,78),(9,57,113,46,133,79),(10,58,114,47,134,80),(11,59,115,48,135,81),(12,60,116,25,136,82),(13,61,117,26,137,83),(14,62,118,27,138,84),(15,63,119,28,139,85),(16,64,120,29,140,86),(17,65,97,30,141,87),(18,66,98,31,142,88),(19,67,99,32,143,89),(20,68,100,33,144,90),(21,69,101,34,121,91),(22,70,102,35,122,92),(23,71,103,36,123,93),(24,72,104,37,124,94)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)])

C6×C24 is a maximal subgroup of
C24.94D6  C12.30Dic6  C24⋊Dic3  C6.4Dic12  C242Dic3  C241Dic3  C12.59D12  C12.60D12  C62.84D4  C24.95D6  C24.78D6

144 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 6A ··· 6X 8A ··· 8H 12A ··· 12AF 24A ··· 24BL order 1 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 kernel C6×C24 C3×C24 C6×C12 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 2 1 8 2 2 16 8 8 16 16 64

Matrix representation of C6×C24 in GL2(𝔽73) generated by

 65 0 0 72
,
 10 0 0 3
G:=sub<GL(2,GF(73))| [65,0,0,72],[10,0,0,3] >;

C6×C24 in GAP, Magma, Sage, TeX

C_6\times C_{24}
% in TeX

G:=Group("C6xC24");
// GroupNames label

G:=SmallGroup(144,104);
// by ID

G=gap.SmallGroup(144,104);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-2,216,88]);
// Polycyclic

G:=Group<a,b|a^6=b^24=1,a*b=b*a>;
// generators/relations

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