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G = C325Q16order 144 = 24·32

2nd semidirect product of C32 and Q16 acting via Q16/C8=C2

metabelian, supersoluble, monomial

Aliases: C24.3S3, C6.9D12, C325Q16, C31Dic12, C12.48D6, C8.(C3⋊S3), (C3×C24).1C2, (C3×C6).24D4, C2.5(C12⋊S3), C324Q8.1C2, (C3×C12).34C22, C4.10(C2×C3⋊S3), SmallGroup(144,89)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C325Q16
C1C3C32C3×C6C3×C12C324Q8 — C325Q16
C32C3×C6C3×C12 — C325Q16
C1C2C4C8

Generators and relations for C325Q16
 G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 178 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C3 [×4], C4, C4 [×2], C6 [×4], C8, Q8 [×2], C32, Dic3 [×8], C12 [×4], Q16, C3×C6, C24 [×4], Dic6 [×8], C3⋊Dic3 [×2], C3×C12, Dic12 [×4], C3×C24, C324Q8 [×2], C325Q16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], Q16, C3⋊S3, D12 [×4], C2×C3⋊S3, Dic12 [×4], C12⋊S3, C325Q16

Smallest permutation representation of C325Q16
Regular action on 144 points
Generators in S144
(1 68 82)(2 69 83)(3 70 84)(4 71 85)(5 72 86)(6 65 87)(7 66 88)(8 67 81)(9 35 56)(10 36 49)(11 37 50)(12 38 51)(13 39 52)(14 40 53)(15 33 54)(16 34 55)(17 59 95)(18 60 96)(19 61 89)(20 62 90)(21 63 91)(22 64 92)(23 57 93)(24 58 94)(25 138 41)(26 139 42)(27 140 43)(28 141 44)(29 142 45)(30 143 46)(31 144 47)(32 137 48)(73 108 132)(74 109 133)(75 110 134)(76 111 135)(77 112 136)(78 105 129)(79 106 130)(80 107 131)(97 124 117)(98 125 118)(99 126 119)(100 127 120)(101 128 113)(102 121 114)(103 122 115)(104 123 116)
(1 114 134)(2 115 135)(3 116 136)(4 117 129)(5 118 130)(6 119 131)(7 120 132)(8 113 133)(9 47 24)(10 48 17)(11 41 18)(12 42 19)(13 43 20)(14 44 21)(15 45 22)(16 46 23)(25 60 37)(26 61 38)(27 62 39)(28 63 40)(29 64 33)(30 57 34)(31 58 35)(32 59 36)(49 137 95)(50 138 96)(51 139 89)(52 140 90)(53 141 91)(54 142 92)(55 143 93)(56 144 94)(65 99 80)(66 100 73)(67 101 74)(68 102 75)(69 103 76)(70 104 77)(71 97 78)(72 98 79)(81 128 109)(82 121 110)(83 122 111)(84 123 112)(85 124 105)(86 125 106)(87 126 107)(88 127 108)
(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 47 5 43)(2 46 6 42)(3 45 7 41)(4 44 8 48)(9 118 13 114)(10 117 14 113)(11 116 15 120)(12 115 16 119)(17 129 21 133)(18 136 22 132)(19 135 23 131)(20 134 24 130)(25 84 29 88)(26 83 30 87)(27 82 31 86)(28 81 32 85)(33 127 37 123)(34 126 38 122)(35 125 39 121)(36 124 40 128)(49 97 53 101)(50 104 54 100)(51 103 55 99)(52 102 56 98)(57 107 61 111)(58 106 62 110)(59 105 63 109)(60 112 64 108)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)(73 96 77 92)(74 95 78 91)(75 94 79 90)(76 93 80 89)

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

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

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

C325Q16 is a maximal subgroup of
C323SD32  C323Q32  C6.D24  C325Q32  C328SD32  C327Q32  S3×Dic12  C24.3D6  D247S3  C24.78D6  C24.5D6  C24.26D6  C24.32D6  Q16×C3⋊S3  He34Q16  C24.D9  C338Q16  C3312Q16
C325Q16 is a maximal quotient of
C6.4Dic12  C241Dic3  C24.D9  He35Q16  C338Q16  C3312Q16

39 conjugacy classes

class 1  2 3A3B3C3D4A4B4C6A6B6C6D8A8B12A···12H24A···24P
order12333344466668812···1224···24
size112222236362222222···22···2

39 irreducible representations

dim111222222
type++++++-+-
imageC1C2C2S3D4D6Q16D12Dic12
kernelC325Q16C3×C24C324Q8C24C3×C6C12C32C6C3
# reps1124142816

Matrix representation of C325Q16 in GL4(𝔽73) generated by

727200
1000
0010
0001
,
0100
727200
0001
007272
,
665900
14700
00235
006818
,
506800
182300
00220
001871
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,72,0,0,0,0,0,72,0,0,1,72],[66,14,0,0,59,7,0,0,0,0,23,68,0,0,5,18],[50,18,0,0,68,23,0,0,0,0,2,18,0,0,20,71] >;

C325Q16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_5Q_{16}
% in TeX

G:=Group("C3^2:5Q16");
// GroupNames label

G:=SmallGroup(144,89);
// by ID

G=gap.SmallGroup(144,89);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,79,218,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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