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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 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×14], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×22], Q8 [×8], Q8 [×12], C23, C42 [×4], C42 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×2], Q16 [×16], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×12], C2×Q8 [×6], C4×C8 [×4], Q8⋊C4 [×8], C2.D8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×Q8 [×8], C4×Q8 [×4], C22×C8 [×2], C2×Q16 [×12], C22×Q8 [×2], C2×C4×C8, C2×Q8⋊C4 [×2], C2×C2.D8, C4×Q16 [×8], C2×C4×Q8 [×2], C22×Q16, C2×C4×Q16
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], Q16 [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×Q16 [×6], C4○D8 [×2], C23×C4, C22×D4, C2×C4○D4, C4×Q16 [×4], 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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