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## G = C32⋊8SD32order 288 = 25·32

### 2nd semidirect product of C32 and SD32 acting via SD32/D8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C32⋊8SD32
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊5Q16 — C32⋊8SD32
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊8SD32
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for C328SD32
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd=c7 >

Subgroups: 312 in 78 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, C6 [×4], C6 [×4], C8, D4, Q8, C32, Dic3 [×4], C12 [×4], C2×C6 [×4], C16, D8, Q16, C3×C6, C3×C6, C24 [×4], Dic6 [×4], C3×D4 [×4], SD32, C3⋊Dic3, C3×C12, C62, C3⋊C16 [×4], Dic12 [×4], C3×D8 [×4], C3×C24, C324Q8, D4×C32, D8.S3 [×4], C24.S3, C325Q16, C32×D8, C328SD32
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, C3⋊D4 [×4], SD32, C2×C3⋊S3, D4⋊S3 [×4], C327D4, D8.S3 [×4], C327D8, C328SD32

Smallest permutation representation of C328SD32
On 144 points
Generators in S144
(1 87 126)(2 127 88)(3 89 128)(4 113 90)(5 91 114)(6 115 92)(7 93 116)(8 117 94)(9 95 118)(10 119 96)(11 81 120)(12 121 82)(13 83 122)(14 123 84)(15 85 124)(16 125 86)(17 143 69)(18 70 144)(19 129 71)(20 72 130)(21 131 73)(22 74 132)(23 133 75)(24 76 134)(25 135 77)(26 78 136)(27 137 79)(28 80 138)(29 139 65)(30 66 140)(31 141 67)(32 68 142)(33 56 104)(34 105 57)(35 58 106)(36 107 59)(37 60 108)(38 109 61)(39 62 110)(40 111 63)(41 64 112)(42 97 49)(43 50 98)(44 99 51)(45 52 100)(46 101 53)(47 54 102)(48 103 55)
(1 17 110)(2 111 18)(3 19 112)(4 97 20)(5 21 98)(6 99 22)(7 23 100)(8 101 24)(9 25 102)(10 103 26)(11 27 104)(12 105 28)(13 29 106)(14 107 30)(15 31 108)(16 109 32)(33 81 137)(34 138 82)(35 83 139)(36 140 84)(37 85 141)(38 142 86)(39 87 143)(40 144 88)(41 89 129)(42 130 90)(43 91 131)(44 132 92)(45 93 133)(46 134 94)(47 95 135)(48 136 96)(49 72 113)(50 114 73)(51 74 115)(52 116 75)(53 76 117)(54 118 77)(55 78 119)(56 120 79)(57 80 121)(58 122 65)(59 66 123)(60 124 67)(61 68 125)(62 126 69)(63 70 127)(64 128 71)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 45)(34 36)(35 43)(37 41)(38 48)(40 46)(42 44)(49 51)(50 58)(52 56)(53 63)(55 61)(57 59)(60 64)(65 73)(66 80)(67 71)(68 78)(70 76)(72 74)(75 79)(81 93)(82 84)(83 91)(85 89)(86 96)(88 94)(90 92)(97 99)(98 106)(100 104)(101 111)(103 109)(105 107)(108 112)(113 115)(114 122)(116 120)(117 127)(119 125)(121 123)(124 128)(129 141)(130 132)(131 139)(133 137)(134 144)(136 142)(138 140)

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

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

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

39 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E ··· 6L 8A 8B 12A 12B 12C 12D 16A 16B 16C 16D 24A ··· 24H order 1 2 2 3 3 3 3 4 4 6 6 6 6 6 ··· 6 8 8 12 12 12 12 16 16 16 16 24 ··· 24 size 1 1 8 2 2 2 2 2 72 2 2 2 2 8 ··· 8 2 2 4 4 4 4 18 18 18 18 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 S3 D4 D6 D8 C3⋊D4 SD32 D4⋊S3 D8.S3 kernel C32⋊8SD32 C24.S3 C32⋊5Q16 C32×D8 C3×D8 C3×C12 C24 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 4 1 4 2 8 4 4 8

Matrix representation of C328SD32 in GL6(𝔽97)

 1 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 61 0 0 0 0 0 0 35
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 1 0 0 0 0 96 0 0 0 0 0 0 0 61 0 0 0 0 0 0 35
,
 10 44 0 0 0 0 53 10 0 0 0 0 0 0 64 83 0 0 0 0 50 33 0 0 0 0 0 0 0 72 0 0 0 0 66 0
,
 1 0 0 0 0 0 0 96 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 96

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,61,0,0,0,0,0,0,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,61,0,0,0,0,0,0,35],[10,53,0,0,0,0,44,10,0,0,0,0,0,0,64,50,0,0,0,0,83,33,0,0,0,0,0,0,0,66,0,0,0,0,72,0],[1,0,0,0,0,0,0,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96] >;

C328SD32 in GAP, Magma, Sage, TeX

C_3^2\rtimes_8{\rm SD}_{32}
% in TeX

G:=Group("C3^2:8SD32");
// GroupNames label

G:=SmallGroup(288,302);
// by ID

G=gap.SmallGroup(288,302);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,254,135,142,675,346,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,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=c^7>;
// generators/relations

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