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G = C32×C3⋊Q16order 432 = 24·33

Direct product of C32 and C3⋊Q16

direct product, metabelian, supersoluble, monomial

Aliases: C32×C3⋊Q16, C3313Q16, C12.4C62, C12.60(S3×C6), C32(C32×Q16), C329(C3×Q16), (C3×C12).187D6, Dic6.2(C3×C6), (C3×Dic6).7C6, C6.10(D4×C32), (C32×C6).69D4, Q8.3(S3×C32), (Q8×C33).1C2, (Q8×C32).22C6, (Q8×C32).28S3, (C32×Dic6).5C2, (C32×C12).26C22, C3⋊C8.(C3×C6), C4.4(S3×C3×C6), (C3×C3⋊C8).4C6, (C3×C6).65(C3×D4), C6.56(C3×C3⋊D4), (C32×C3⋊C8).3C2, (C3×C12).45(C2×C6), (C3×Q8).13(C3×C6), (C3×Q8).36(C3×S3), C2.7(C32×C3⋊D4), (C3×C6).125(C3⋊D4), SmallGroup(432,478)

Series: Derived Chief Lower central Upper central

C1C12 — C32×C3⋊Q16
C1C3C6C12C3×C12C32×C12C32×Dic6 — C32×C3⋊Q16
C3C6C12 — C32×C3⋊Q16
C1C3×C6C3×C12Q8×C32

Generators and relations for C32×C3⋊Q16
 G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 288 in 156 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C32×C6, C3×C3⋊C8, C3×C24, C3×Dic6, Q8×C32, Q8×C32, Q8×C32, C32×Dic3, C32×C12, C32×C12, C3×C3⋊Q16, C32×Q16, C32×C3⋊C8, C32×Dic6, Q8×C33, C32×C3⋊Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, Q16, C3×S3, C3×C6, C3⋊D4, C3×D4, S3×C6, C62, C3⋊Q16, C3×Q16, S3×C32, C3×C3⋊D4, D4×C32, S3×C3×C6, C3×C3⋊Q16, C32×Q16, C32×C3⋊D4, C32×C3⋊Q16

Smallest permutation representation of C32×C3⋊Q16
On 144 points
Generators in S144
(1 118 9)(2 119 10)(3 120 11)(4 113 12)(5 114 13)(6 115 14)(7 116 15)(8 117 16)(17 37 28)(18 38 29)(19 39 30)(20 40 31)(21 33 32)(22 34 25)(23 35 26)(24 36 27)(41 106 49)(42 107 50)(43 108 51)(44 109 52)(45 110 53)(46 111 54)(47 112 55)(48 105 56)(57 129 124)(58 130 125)(59 131 126)(60 132 127)(61 133 128)(62 134 121)(63 135 122)(64 136 123)(65 100 90)(66 101 91)(67 102 92)(68 103 93)(69 104 94)(70 97 95)(71 98 96)(72 99 89)(73 143 81)(74 144 82)(75 137 83)(76 138 84)(77 139 85)(78 140 86)(79 141 87)(80 142 88)
(1 44 139)(2 45 140)(3 46 141)(4 47 142)(5 48 143)(6 41 144)(7 42 137)(8 43 138)(9 52 77)(10 53 78)(11 54 79)(12 55 80)(13 56 73)(14 49 74)(15 50 75)(16 51 76)(17 60 96)(18 61 89)(19 62 90)(20 63 91)(21 64 92)(22 57 93)(23 58 94)(24 59 95)(25 124 103)(26 125 104)(27 126 97)(28 127 98)(29 128 99)(30 121 100)(31 122 101)(32 123 102)(33 136 67)(34 129 68)(35 130 69)(36 131 70)(37 132 71)(38 133 72)(39 134 65)(40 135 66)(81 114 105)(82 115 106)(83 116 107)(84 117 108)(85 118 109)(86 119 110)(87 120 111)(88 113 112)
(1 44 139)(2 140 45)(3 46 141)(4 142 47)(5 48 143)(6 144 41)(7 42 137)(8 138 43)(9 52 77)(10 78 53)(11 54 79)(12 80 55)(13 56 73)(14 74 49)(15 50 75)(16 76 51)(17 60 96)(18 89 61)(19 62 90)(20 91 63)(21 64 92)(22 93 57)(23 58 94)(24 95 59)(25 103 124)(26 125 104)(27 97 126)(28 127 98)(29 99 128)(30 121 100)(31 101 122)(32 123 102)(33 136 67)(34 68 129)(35 130 69)(36 70 131)(37 132 71)(38 72 133)(39 134 65)(40 66 135)(81 114 105)(82 106 115)(83 116 107)(84 108 117)(85 118 109)(86 110 119)(87 120 111)(88 112 113)
(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 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 46 21 42)(18 45 22 41)(19 44 23 48)(20 43 24 47)(25 49 29 53)(26 56 30 52)(27 55 31 51)(28 54 32 50)(33 107 37 111)(34 106 38 110)(35 105 39 109)(36 112 40 108)(57 144 61 140)(58 143 62 139)(59 142 63 138)(60 141 64 137)(65 118 69 114)(66 117 70 113)(67 116 71 120)(68 115 72 119)(73 121 77 125)(74 128 78 124)(75 127 79 123)(76 126 80 122)(81 134 85 130)(82 133 86 129)(83 132 87 136)(84 131 88 135)

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

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

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

108 conjugacy classes

class 1  2 3A···3H3I···3Q4A4B4C6A···6H6I···6Q8A8B12A···12H12I···12AQ12AR···12AY24A···24P
order123···33···34446···66···68812···1212···1212···1224···24
size111···12···224121···12···2662···24···412···126···6

108 irreducible representations

dim11111111222222222244
type+++++++--
imageC1C2C2C2C3C6C6C6S3D4D6Q16C3×S3C3⋊D4C3×D4S3×C6C3×Q16C3×C3⋊D4C3⋊Q16C3×C3⋊Q16
kernelC32×C3⋊Q16C32×C3⋊C8C32×Dic6Q8×C33C3×C3⋊Q16C3×C3⋊C8C3×Dic6Q8×C32Q8×C32C32×C6C3×C12C33C3×Q8C3×C6C3×C6C12C32C6C32C3
# reps1111888811128288161618

Matrix representation of C32×C3⋊Q16 in GL4(𝔽73) generated by

1000
0100
0080
0008
,
8000
0800
0010
0001
,
8000
526400
0010
0001
,
511000
322200
003232
00570
,
72000
07200
00659
006067
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[8,52,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[51,32,0,0,10,22,0,0,0,0,32,57,0,0,32,0],[72,0,0,0,0,72,0,0,0,0,6,60,0,0,59,67] >;

C32×C3⋊Q16 in GAP, Magma, Sage, TeX

C_3^2\times C_3\rtimes Q_{16}
% in TeX

G:=Group("C3^2xC3:Q16");
// GroupNames label

G:=SmallGroup(432,478);
// by ID

G=gap.SmallGroup(432,478);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,512,3784,1901,102,14118]);
// Polycyclic

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

׿
×
𝔽