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G = C23×Q16order 128 = 27

Direct product of C23 and Q16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C23×Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C23×Q16
 Lower central C1 — C2 — C4 — C23×Q16
 Upper central C1 — C24 — C23×C4 — C23×Q16
 Jennings C1 — C2 — C2 — C4 — C23×Q16

Generators and relations for C23×Q16
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 988 in 732 conjugacy classes, 476 normal (7 characteristic)
C1, C2, C2 [×14], C4, C4 [×7], C4 [×16], C22 [×35], C8 [×8], C2×C4 [×28], C2×C4 [×56], Q8 [×16], Q8 [×56], C23 [×15], C2×C8 [×28], Q16 [×64], C22×C4 [×14], C22×C4 [×28], C2×Q8 [×56], C2×Q8 [×84], C24, C22×C8 [×14], C2×Q16 [×112], C23×C4, C23×C4 [×2], C22×Q8 [×28], C22×Q8 [×14], C23×C8, C22×Q16 [×28], Q8×C23 [×2], C23×Q16
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], Q16 [×8], C2×D4 [×28], C24 [×31], C2×Q16 [×28], C22×D4 [×14], C25, C22×Q16 [×14], D4×C23, C23×Q16

Smallest permutation representation of C23×Q16
Regular action on 128 points
Generators in S128
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 97)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(25 69)(26 70)(27 71)(28 72)(29 65)(30 66)(31 67)(32 68)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 113)(41 128)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(73 92)(74 93)(75 94)(76 95)(77 96)(78 89)(79 90)(80 91)(81 111)(82 112)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)
(1 48)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 70)(34 71)(35 72)(36 65)(37 66)(38 67)(39 68)(40 69)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 81)(57 77)(58 78)(59 79)(60 80)(61 73)(62 74)(63 75)(64 76)(97 126)(98 127)(99 128)(100 121)(101 122)(102 123)(103 124)(104 125)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 113)(8 114)(9 87)(10 88)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(49 92)(50 93)(51 94)(52 95)(53 96)(54 89)(55 90)(56 91)(57 108)(58 109)(59 110)(60 111)(61 112)(62 105)(63 106)(64 107)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)
(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)
(1 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 29 21 25)(18 28 22 32)(19 27 23 31)(20 26 24 30)(33 82 37 86)(34 81 38 85)(35 88 39 84)(36 87 40 83)(41 79 45 75)(42 78 46 74)(43 77 47 73)(44 76 48 80)(49 66 53 70)(50 65 54 69)(51 72 55 68)(52 71 56 67)(89 125 93 121)(90 124 94 128)(91 123 95 127)(92 122 96 126)(105 117 109 113)(106 116 110 120)(107 115 111 119)(108 114 112 118)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4H 4I ··· 4X 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + + + - image C1 C2 C2 C2 D4 D4 Q16 kernel C23×Q16 C23×C8 C22×Q16 Q8×C23 C22×C4 C24 C23 # reps 1 1 28 2 7 1 16

Matrix representation of C23×Q16 in GL7(𝔽17)

 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 7 13 0 0 0 0 0 4 10 0 0 0 0 0 0 0 10 4 0 0 0 0 0 13 7 0 0 0 0 0 0 0 3 14 0 0 0 0 0 3 3
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 0 0 10 16 0 0 0 0 0 16 7

G:=sub<GL(7,GF(17))| [16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,7,4,0,0,0,0,0,13,10,0,0,0,0,0,0,0,10,13,0,0,0,0,0,4,7,0,0,0,0,0,0,0,3,3,0,0,0,0,0,14,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,10,16,0,0,0,0,0,16,7] >;

C23×Q16 in GAP, Magma, Sage, TeX

C_2^3\times Q_{16}
% in TeX

G:=Group("C2^3xQ16");
// GroupNames label

G:=SmallGroup(128,2308);
// by ID

G=gap.SmallGroup(128,2308);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,456,4037,2028,124]);
// Polycyclic

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

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