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G = C327Q16order 144 = 24·32

2nd semidirect product of C32 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial

Aliases: C327Q16, C12.19D6, (C3×C6).37D4, (C3×Q8).9S3, C33(C3⋊Q16), Q8.2(C3⋊S3), C6.25(C3⋊D4), C324C8.2C2, (Q8×C32).2C2, C324Q8.3C2, (C3×C12).15C22, C2.7(C327D4), C4.4(C2×C3⋊S3), SmallGroup(144,99)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C327Q16
C1C3C32C3×C6C3×C12C324Q8 — C327Q16
C32C3×C6C3×C12 — C327Q16
C1C2C4Q8

Generators and relations for C327Q16
 G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 146 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C3 [×4], C4, C4 [×2], C6 [×4], C8, Q8, Q8, C32, Dic3 [×4], C12 [×4], C12 [×4], Q16, C3×C6, C3⋊C8 [×4], Dic6 [×4], C3×Q8 [×4], C3⋊Dic3, C3×C12, C3×C12, C3⋊Q16 [×4], C324C8, C324Q8, Q8×C32, C327Q16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], Q16, C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, C3⋊Q16 [×4], C327D4, C327Q16

Character table of C327Q16

 class 123A3B3C3D4A4B4C6A6B6C6D8A8B12A12B12C12D12E12F12G12H12I12J12K12L
 size 112222243622221818444444444444
ρ1111111111111111111111111111    trivial
ρ211111111-11111-1-1111111111111    linear of order 2
ρ31111111-1-11111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ41111111-111111-1-11-1-1-1-1-1-1-1-1111    linear of order 2
ρ522-1-12-1220-12-1-100222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ622-1-1-12220-1-12-100-1-1-12-1-12-1-12-1-1    orthogonal lifted from S3
ρ722-1-1-122-20-1-12-100-111-211-2112-1-1    orthogonal lifted from D6
ρ8222-1-1-12202-1-1-100-1-1-1-1-12-12-1-1-12    orthogonal lifted from S3
ρ922-1-12-12-20-12-1-1002-2-2111111-1-1-1    orthogonal lifted from D6
ρ10222-1-1-12-202-1-1-100-11111-21-21-1-12    orthogonal lifted from D6
ρ1122-12-1-12-20-1-1-1200-1111-2111-2-12-1    orthogonal lifted from D6
ρ12222222-200222200-200000000-2-2-2    orthogonal lifted from D4
ρ1322-12-1-1220-1-1-1200-1-1-1-12-1-1-12-12-1    orthogonal lifted from S3
ρ142-22222000-2-2-2-2-22000000000000    symplectic lifted from Q16, Schur index 2
ρ152-22222000-2-2-2-22-2000000000000    symplectic lifted from Q16, Schur index 2
ρ1622-12-1-1-200-1-1-12001--3-3-30--3--3-301-21    complex lifted from C3⋊D4
ρ17222-1-1-1-2002-1-1-1001-3--3-3--30--30-311-2    complex lifted from C3⋊D4
ρ1822-1-12-1-200-12-1-100-200--3--3--3-3-3-3111    complex lifted from C3⋊D4
ρ1922-1-1-12-200-1-12-1001--3-30--3-30--3-3-211    complex lifted from C3⋊D4
ρ2022-12-1-1-200-1-1-12001-3--3--30-3-3--301-21    complex lifted from C3⋊D4
ρ2122-1-1-12-200-1-12-1001-3--30-3--30-3--3-211    complex lifted from C3⋊D4
ρ22222-1-1-1-2002-1-1-1001--3-3--3-30-30--311-2    complex lifted from C3⋊D4
ρ2322-1-12-1-200-12-1-100-200-3-3-3--3--3--3111    complex lifted from C3⋊D4
ρ244-4-24-2-2000222-400000000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ254-4-2-24-20002-42200000000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ264-44-2-2-2000-422200000000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ274-4-2-2-2400022-4200000000000000    symplectic lifted from C3⋊Q16, Schur index 2

Smallest permutation representation of C327Q16
Regular action on 144 points
Generators in S144
(1 72 11)(2 12 65)(3 66 13)(4 14 67)(5 68 15)(6 16 69)(7 70 9)(8 10 71)(17 123 107)(18 108 124)(19 125 109)(20 110 126)(21 127 111)(22 112 128)(23 121 105)(24 106 122)(25 44 91)(26 92 45)(27 46 93)(28 94 47)(29 48 95)(30 96 41)(31 42 89)(32 90 43)(33 113 79)(34 80 114)(35 115 73)(36 74 116)(37 117 75)(38 76 118)(39 119 77)(40 78 120)(49 144 64)(50 57 137)(51 138 58)(52 59 139)(53 140 60)(54 61 141)(55 142 62)(56 63 143)(81 134 101)(82 102 135)(83 136 103)(84 104 129)(85 130 97)(86 98 131)(87 132 99)(88 100 133)
(1 106 31)(2 32 107)(3 108 25)(4 26 109)(5 110 27)(6 28 111)(7 112 29)(8 30 105)(9 22 95)(10 96 23)(11 24 89)(12 90 17)(13 18 91)(14 92 19)(15 20 93)(16 94 21)(33 137 102)(34 103 138)(35 139 104)(36 97 140)(37 141 98)(38 99 142)(39 143 100)(40 101 144)(41 121 71)(42 72 122)(43 123 65)(44 66 124)(45 125 67)(46 68 126)(47 127 69)(48 70 128)(49 120 134)(50 135 113)(51 114 136)(52 129 115)(53 116 130)(54 131 117)(55 118 132)(56 133 119)(57 82 79)(58 80 83)(59 84 73)(60 74 85)(61 86 75)(62 76 87)(63 88 77)(64 78 81)
(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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 56 13 52)(10 55 14 51)(11 54 15 50)(12 53 16 49)(17 130 21 134)(18 129 22 133)(19 136 23 132)(20 135 24 131)(25 73 29 77)(26 80 30 76)(27 79 31 75)(28 78 32 74)(33 42 37 46)(34 41 38 45)(35 48 39 44)(36 47 40 43)(65 140 69 144)(66 139 70 143)(67 138 71 142)(68 137 72 141)(81 107 85 111)(82 106 86 110)(83 105 87 109)(84 112 88 108)(89 117 93 113)(90 116 94 120)(91 115 95 119)(92 114 96 118)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)

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

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

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

C327Q16 is a maximal subgroup of
S3×C3⋊Q16  D12.11D6  D12.24D6  D12.12D6  C24.32D6  C24.40D6  Q16×C3⋊S3  C24.35D6  C62.134D4  C62.74D4  C62.75D4  He36Q16  C36.19D6  C32.3CSU2(𝔽3)  C336Q16  C337Q16  C3315Q16  C324CSU2(𝔽3)
C327Q16 is a maximal quotient of
C12.9Dic6  C62.114D4  C62.117D4  C36.19D6  He37Q16  C336Q16  C337Q16  C3315Q16

Matrix representation of C327Q16 in GL6(𝔽73)

010000
72720000
001000
000100
000010
000001
,
010000
72720000
008000
00426400
000010
000001
,
41510000
10320000
0075800
00526600
00004138
0000480
,
7200000
0720000
001000
00357200
00007159
0000162

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,8,42,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[41,10,0,0,0,0,51,32,0,0,0,0,0,0,7,52,0,0,0,0,58,66,0,0,0,0,0,0,41,48,0,0,0,0,38,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,35,0,0,0,0,0,72,0,0,0,0,0,0,71,16,0,0,0,0,59,2] >;

C327Q16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_7Q_{16}
% in TeX

G:=Group("C3^2:7Q16");
// GroupNames label

G:=SmallGroup(144,99);
// by ID

G=gap.SmallGroup(144,99);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,55,218,116,50,964,3461]);
// Polycyclic

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

Export

Character table of C327Q16 in TeX

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