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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, C4, C4, C4, C22, C8, C2×C4, C2×C4, Q8, Q8, C23, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C22×C8, C2×Q16, C23×C4, C23×C4, C22×Q8, C22×Q8, C23×C8, C22×Q16, Q8×C23, C23×Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C2×Q16, C22×D4, C25, C22×Q16, 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 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(41 128)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 111)(74 112)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)
(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 71)(34 72)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(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 79)(10 80)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 84)(18 85)(19 86)(20 87)(21 88)(22 81)(23 82)(24 83)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)(33 125)(34 126)(35 127)(36 128)(37 121)(38 122)(39 123)(40 124)(49 110)(50 111)(51 112)(52 105)(53 106)(54 107)(55 108)(56 109)(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 89)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 97)
(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 50 5 54)(2 49 6 53)(3 56 7 52)(4 55 8 51)(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 60 37 64)(34 59 38 63)(35 58 39 62)(36 57 40 61)(41 81 45 85)(42 88 46 84)(43 87 47 83)(44 86 48 82)(65 73 69 77)(66 80 70 76)(67 79 71 75)(68 78 72 74)(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)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,128)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,111)(74,112)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (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)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,80)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(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)(9,79)(10,80)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,84)(18,85)(19,86)(20,87)(21,88)(22,81)(23,82)(24,83)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45)(33,125)(34,126)(35,127)(36,128)(37,121)(38,122)(39,123)(40,124)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,89)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,97), (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), (1,50,5,54)(2,49,6,53)(3,56,7,52)(4,55,8,51)(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)(33,60,37,64)(34,59,38,63)(35,58,39,62)(36,57,40,61)(41,81,45,85)(42,88,46,84)(43,87,47,83)(44,86,48,82)(65,73,69,77)(66,80,70,76)(67,79,71,75)(68,78,72,74)(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)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,128)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,111)(74,112)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (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)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,80)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(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)(9,79)(10,80)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,84)(18,85)(19,86)(20,87)(21,88)(22,81)(23,82)(24,83)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45)(33,125)(34,126)(35,127)(36,128)(37,121)(38,122)(39,123)(40,124)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,89)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,97), (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), (1,50,5,54)(2,49,6,53)(3,56,7,52)(4,55,8,51)(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)(33,60,37,64)(34,59,38,63)(35,58,39,62)(36,57,40,61)(41,81,45,85)(42,88,46,84)(43,87,47,83)(44,86,48,82)(65,73,69,77)(66,80,70,76)(67,79,71,75)(68,78,72,74)(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),(9,56),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(41,128),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,113),(72,114),(73,111),(74,112),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,90),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,89)], [(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),(33,71),(34,72),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,80),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(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),(9,79),(10,80),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,84),(18,85),(19,86),(20,87),(21,88),(22,81),(23,82),(24,83),(25,46),(26,47),(27,48),(28,41),(29,42),(30,43),(31,44),(32,45),(33,125),(34,126),(35,127),(36,128),(37,121),(38,122),(39,123),(40,124),(49,110),(50,111),(51,112),(52,105),(53,106),(54,107),(55,108),(56,109),(57,90),(58,91),(59,92),(60,93),(61,94),(62,95),(63,96),(64,89),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,97)], [(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)], [(1,50,5,54),(2,49,6,53),(3,56,7,52),(4,55,8,51),(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),(33,60,37,64),(34,59,38,63),(35,58,39,62),(36,57,40,61),(41,81,45,85),(42,88,46,84),(43,87,47,83),(44,86,48,82),(65,73,69,77),(66,80,70,76),(67,79,71,75),(68,78,72,74),(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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