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## G = (C2×C4)⋊9Q16order 128 = 27

### 1st semidirect product of C2×C4 and Q16 acting via Q16/Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C4)⋊9Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4×Q8 — (C2×C4)⋊9Q16
 Lower central C1 — C2 — C2×C4 — (C2×C4)⋊9Q16
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊9Q16
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊9Q16

Generators and relations for (C2×C4)⋊9Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=ab-1, bd=db, dcd-1=c-1 >

Subgroups: 308 in 169 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×C8, C2×Q16, C2×Q16, C22×Q8, C22.7C42, C22.4Q16, C23.67C23, C2×Q8⋊C4, C2×C4×Q8, C22×Q16, (C2×C4)⋊9Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, Q16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×Q16, C4○D8, C8.C22, C23.23D4, C4×Q16, Q16⋊C4, C22⋊Q16, D4.7D4, C42Q16, Q8.D4, (C2×C4)⋊9Q16

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 Q16 C4○D4 C4○D8 C8.C22 kernel (C2×C4)⋊9Q16 C22.7C42 C22.4Q16 C23.67C23 C2×Q8⋊C4 C2×C4×Q8 C22×Q16 C2×Q16 C4⋊C4 C22×C4 C2×Q8 C2×C4 C2×C4 C22 C22 # reps 1 1 1 1 2 1 1 8 2 2 4 4 4 4 2

Matrix representation of (C2×C4)⋊9Q16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 10 11 0 0 0 0 14 7 0 0 0 0 0 0 3 3 0 0 0 0 14 3
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 16 0 0 0 0 0 1 0 0 0 0 0 0 1 7 0 0 0 0 7 16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,10,14,0,0,0,0,11,7,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,0,0,0,1,7,0,0,0,0,7,16] >;

(C2×C4)⋊9Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_9Q_{16}
% in TeX

G:=Group("(C2xC4):9Q16");
// GroupNames label

G:=SmallGroup(128,610);
// by ID

G=gap.SmallGroup(128,610);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,1018,521,248,1411,718,172,2028,1027,124]);
// 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,c*b*c^-1=a*b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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