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## G = C2×C4×Q16order 128 = 27

### Direct product of C2×C4 and Q16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C4×Q16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×Q8 — C2×C4×Q16
 Lower central C1 — C2 — C4 — C2×C4×Q16
 Upper central C1 — C22×C4 — C2×C42 — C2×C4×Q16
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4×Q16

Generators and relations for C2×C4×Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 348 in 244 conjugacy classes, 156 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×C8, C2×Q16, C22×Q8, C2×C4×C8, C2×Q8⋊C4, C2×C2.D8, C4×Q16, C2×C4×Q8, C22×Q16, C2×C4×Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, Q16, C22×C4, C2×D4, C4○D4, C24, C4×D4, C2×Q16, C4○D8, C23×C4, C22×D4, C2×C4○D4, C4×Q16, C2×C4×D4, C22×Q16, C2×C4○D8, C2×C4×Q16

Smallest permutation representation of C2×C4×Q16
Regular action on 128 points
Generators in S128
(1 119)(2 120)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 127)(64 128)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(73 102)(74 103)(75 104)(76 97)(77 98)(78 99)(79 100)(80 101)(81 94)(82 95)(83 96)(84 89)(85 90)(86 91)(87 92)(88 93)
(1 56 124 107)(2 49 125 108)(3 50 126 109)(4 51 127 110)(5 52 128 111)(6 53 121 112)(7 54 122 105)(8 55 123 106)(9 58 72 117)(10 59 65 118)(11 60 66 119)(12 61 67 120)(13 62 68 113)(14 63 69 114)(15 64 70 115)(16 57 71 116)(17 96 79 34)(18 89 80 35)(19 90 73 36)(20 91 74 37)(21 92 75 38)(22 93 76 39)(23 94 77 40)(24 95 78 33)(25 44 87 104)(26 45 88 97)(27 46 81 98)(28 47 82 99)(29 48 83 100)(30 41 84 101)(31 42 85 102)(32 43 86 103)
(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)
(1 103 5 99)(2 102 6 98)(3 101 7 97)(4 100 8 104)(9 39 13 35)(10 38 14 34)(11 37 15 33)(12 36 16 40)(17 59 21 63)(18 58 22 62)(19 57 23 61)(20 64 24 60)(25 51 29 55)(26 50 30 54)(27 49 31 53)(28 56 32 52)(41 122 45 126)(42 121 46 125)(43 128 47 124)(44 127 48 123)(65 92 69 96)(66 91 70 95)(67 90 71 94)(68 89 72 93)(73 116 77 120)(74 115 78 119)(75 114 79 118)(76 113 80 117)(81 108 85 112)(82 107 86 111)(83 106 87 110)(84 105 88 109)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 Q16 C4○D4 C4○D8 kernel C2×C4×Q16 C2×C4×C8 C2×Q8⋊C4 C2×C2.D8 C4×Q16 C2×C4×Q8 C22×Q16 C2×Q16 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 1 2 1 8 2 1 16 2 2 8 4 8

Matrix representation of C2×C4×Q16 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 0 11 0 0 3 11
,
 16 0 0 0 0 16 0 0 0 0 4 9 0 0 0 13
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,9,13] >;

C2×C4×Q16 in GAP, Magma, Sage, TeX

C_2\times C_4\times Q_{16}
% in TeX

G:=Group("C2xC4xQ16");
// GroupNames label

G:=SmallGroup(128,1670);
// by ID

G=gap.SmallGroup(128,1670);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,520,2804,1411,172]);
// Polycyclic

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

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