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G = C24.S3order 144 = 24·32

9th non-split extension by C24 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C24.9S3, C324C16, C12.7Dic3, C3⋊(C3⋊C16), C6.3(C3⋊C8), (C3×C6).4C8, C8.2(C3⋊S3), (C3×C24).5C2, (C3×C12).7C4, C2.(C324C8), C4.2(C3⋊Dic3), SmallGroup(144,29)

Series: Derived Chief Lower central Upper central

C1C32 — C24.S3
C1C3C32C3×C6C3×C12C3×C24 — C24.S3
C32 — C24.S3
C1C8

Generators and relations for C24.S3
 G = < a,b,c | a24=b3=1, c2=a9, ab=ba, cac-1=a17, cbc-1=b-1 >

9C16
3C3⋊C16
3C3⋊C16
3C3⋊C16
3C3⋊C16

Smallest permutation representation of C24.S3
Regular action on 144 points
Generators in S144
(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)
(1 58 142)(2 59 143)(3 60 144)(4 61 121)(5 62 122)(6 63 123)(7 64 124)(8 65 125)(9 66 126)(10 67 127)(11 68 128)(12 69 129)(13 70 130)(14 71 131)(15 72 132)(16 49 133)(17 50 134)(18 51 135)(19 52 136)(20 53 137)(21 54 138)(22 55 139)(23 56 140)(24 57 141)(25 79 106)(26 80 107)(27 81 108)(28 82 109)(29 83 110)(30 84 111)(31 85 112)(32 86 113)(33 87 114)(34 88 115)(35 89 116)(36 90 117)(37 91 118)(38 92 119)(39 93 120)(40 94 97)(41 95 98)(42 96 99)(43 73 100)(44 74 101)(45 75 102)(46 76 103)(47 77 104)(48 78 105)
(1 79 10 88 19 73 4 82 13 91 22 76 7 85 16 94)(2 96 11 81 20 90 5 75 14 84 23 93 8 78 17 87)(3 89 12 74 21 83 6 92 15 77 24 86 9 95 18 80)(25 67 34 52 43 61 28 70 37 55 46 64 31 49 40 58)(26 60 35 69 44 54 29 63 38 72 47 57 32 66 41 51)(27 53 36 62 45 71 30 56 39 65 48 50 33 59 42 68)(97 142 106 127 115 136 100 121 109 130 118 139 103 124 112 133)(98 135 107 144 116 129 101 138 110 123 119 132 104 141 113 126)(99 128 108 137 117 122 102 131 111 140 120 125 105 134 114 143)

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

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

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

C24.S3 is a maximal subgroup of
C322C32  S3×C3⋊C16  C24.61D6  C322D16  D24.S3  C322Q32  C16×C3⋊S3  C48⋊S3  C24.94D6  C327D16  C328SD32  C3210SD32  C327Q32  He33C16  C72.S3  C337C16
C24.S3 is a maximal quotient of
C48.S3  C72.S3  He34C16  C337C16

48 conjugacy classes

class 1  2 3A3B3C3D4A4B6A6B6C6D8A8B8C8D12A···12H16A···16H24A···24P
order123333446666888812···1216···1624···24
size11222211222211112···29···92···2

48 irreducible representations

dim111112222
type+++-
imageC1C2C4C8C16S3Dic3C3⋊C8C3⋊C16
kernelC24.S3C3×C24C3×C12C3×C6C32C24C12C6C3
# reps1124844816

Matrix representation of C24.S3 in GL5(𝔽97)

470000
002200
0757500
00011
000960
,
10000
0969600
01000
0009696
00010
,
850000
0281000
0796900
0005389
0003644

G:=sub<GL(5,GF(97))| [47,0,0,0,0,0,0,75,0,0,0,22,75,0,0,0,0,0,1,96,0,0,0,1,0],[1,0,0,0,0,0,96,1,0,0,0,96,0,0,0,0,0,0,96,1,0,0,0,96,0],[85,0,0,0,0,0,28,79,0,0,0,10,69,0,0,0,0,0,53,36,0,0,0,89,44] >;

C24.S3 in GAP, Magma, Sage, TeX

C_{24}.S_3
% in TeX

G:=Group("C24.S3");
// GroupNames label

G:=SmallGroup(144,29);
// by ID

G=gap.SmallGroup(144,29);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,12,31,50,964,3461]);
// Polycyclic

G:=Group<a,b,c|a^24=b^3=1,c^2=a^9,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C24.S3 in TeX

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