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## G = C2×C4⋊Q16order 128 = 27

### Direct product of C2 and C4⋊Q16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4⋊Q16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C2×C4⋊Q16
 Lower central C1 — C2 — C2×C4 — C2×C4⋊Q16
 Upper central C1 — C23 — 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, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 468 in 276 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×24], C23, C42 [×4], C4⋊C4 [×16], C2×C8 [×12], Q16 [×32], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×8], C2×Q8 [×20], C4×C8 [×4], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C2×Q16 [×16], C2×Q16 [×16], C22×Q8 [×4], C2×C4×C8, C4⋊Q16 [×8], C2×C4⋊Q8 [×2], C22×Q16 [×4], C2×C4⋊Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], Q16 [×8], C2×D4 [×18], C24, C41D4 [×4], C2×Q16 [×12], C22×D4 [×3], C4⋊Q16 [×4], C2×C41D4, C22×Q16 [×2], C2×C4⋊Q16

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

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

Matrix representation of C2×C4⋊Q16 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 13 0 0 0 0 8 4
,
 16 0 0 0 0 0 8 0 0 0 0 12 15 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 2 0 0 0 16 16 0 0 0 0 0 16 16 0 0 0 0 1

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,8,0,0,0,0,4],[16,0,0,0,0,0,8,12,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,16,0,0,0,2,16,0,0,0,0,0,16,0,0,0,0,16,1] >;

C2×C4⋊Q16 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4:Q16");
// GroupNames label

G:=SmallGroup(128,1877);
// by ID

G=gap.SmallGroup(128,1877);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,184,2804,172]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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