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## G = (C2×C4)⋊3Q16order 128 = 27

### 2nd semidirect product of C2×C4 and Q16 acting via Q16/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4)⋊3Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×Q8⋊C4 — (C2×C4)⋊3Q16
 Lower central C1 — C2 — C22×C4 — (C2×C4)⋊3Q16
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊3Q16
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊3Q16

Generators and relations for (C2×C4)⋊3Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=c-1 >

Subgroups: 288 in 144 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C2×Q16, C22×Q8, C22.4Q16, C23.67C23, C2×Q8⋊C4, C2×C4⋊C8, C22×Q16, (C2×C4)⋊3Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×Q16, C4○D8, C8.C22, C23.10D4, C42Q16, Q8.D4, C8.18D4, C8.D4, (C2×C4)⋊3Q16

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

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

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

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

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 2 2 2 2 2 2 4 type + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D4 Q16 C4○D4 C4○D8 C8.C22 kernel (C2×C4)⋊3Q16 C22.4Q16 C23.67C23 C2×Q8⋊C4 C2×C4⋊C8 C22×Q16 C2×C8 C22×C4 C2×Q8 C2×C4 C2×C4 C22 C22 # reps 1 1 2 2 1 1 2 2 4 4 6 4 2

Matrix representation of (C2×C4)⋊3Q16 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 7 1 0 0 0 0 1 10
,
 9 0 0 0 0 0 5 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 7 0 0 0 0 7 1
,
 16 15 0 0 0 0 1 1 0 0 0 0 0 0 11 1 0 0 0 0 16 6 0 0 0 0 0 0 0 4 0 0 0 0 13 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,1,0,0,0,0,1,10],[9,5,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,7,0,0,0,0,7,1],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,11,16,0,0,0,0,1,6,0,0,0,0,0,0,0,13,0,0,0,0,4,0] >;

(C2×C4)⋊3Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3Q_{16}
% in TeX

G:=Group("(C2xC4):3Q16");
// GroupNames label

G:=SmallGroup(128,788);
// by ID

G=gap.SmallGroup(128,788);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,58,2019,1018,248,2804,1411,172,4037,2028,124]);
// Polycyclic

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

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