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## G = (C2×Q16)⋊10C4order 128 = 27

### 6th semidirect product of C2×Q16 and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×Q16)⋊10C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — (C2×Q16)⋊10C4
 Lower central C1 — C2 — C2×C4 — (C2×Q16)⋊10C4
 Upper central C1 — C23 — C2×C42 — (C2×Q16)⋊10C4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×Q16)⋊10C4

Generators and relations for (C2×Q16)⋊10C4
G = < a,b,c,d | a2=b8=d4=1, c2=b4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b3, dcd-1=ab4c >

Subgroups: 292 in 152 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C8⋊C4, Q8⋊C4, C4.Q8, C2×C42, C2×C4⋊C4, C22×C8, C2×Q16, C2×Q16, C22×Q8, C23.67C23, C2×C8⋊C4, C2×Q8⋊C4, C2×C4.Q8, C22×Q16, (C2×Q16)⋊10C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C41D4, C8.C22, C24.3C22, Q16⋊C4, C8.D4, C8.2D4, (C2×Q16)⋊10C4

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

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8.C22 kernel (C2×Q16)⋊10C4 C23.67C23 C2×C8⋊C4 C2×Q8⋊C4 C2×C4.Q8 C22×Q16 C2×Q16 C2×C8 C22×C4 C2×C4 C22 # reps 1 2 1 2 1 1 8 6 2 4 4

Matrix representation of (C2×Q16)⋊10C4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 0 4 0 4 15 0 0 0 0 8 9 10 13
,
 4 11 0 0 0 0 0 0 11 13 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 3 0 4 7 0 0 0 0 13 0 16 2 0 0 0 0 1 5 0 0 0 0 0 0 15 2 8 14
,
 7 16 0 0 0 0 0 0 16 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 4 0 4 15 0 0 0 0 16 0 0 0 0 0 0 0 15 9 2 0

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,4,8,0,0,0,0,13,0,0,9,0,0,0,0,0,0,4,10,0,0,0,0,0,0,15,13],[4,11,0,0,0,0,0,0,11,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,13,1,15,0,0,0,0,0,0,5,2,0,0,0,0,4,16,0,8,0,0,0,0,7,2,0,14],[7,16,0,0,0,0,0,0,16,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,16,15,0,0,0,0,0,0,0,9,0,0,0,0,1,4,0,2,0,0,0,0,0,15,0,0] >;

(C2×Q16)⋊10C4 in GAP, Magma, Sage, TeX

(C_2\times Q_{16})\rtimes_{10}C_4
% in TeX

G:=Group("(C2xQ16):10C4");
// GroupNames label

G:=SmallGroup(128,703);
// by ID

G=gap.SmallGroup(128,703);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^3,d*c*d^-1=a*b^4*c>;
// generators/relations

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