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## G = C2×C22.4Q16order 128 = 27

### Direct product of C2 and C22.4Q16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C22.4Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C2×C22.4Q16
 Lower central C1 — C2 — C4 — C2×C22.4Q16
 Upper central C1 — C24 — C23×C4 — C2×C22.4Q16
 Jennings C1 — C2 — C2 — C22×C4 — C2×C22.4Q16

Generators and relations for C2×C22.4Q16
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=cd4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 436 in 256 conjugacy classes, 156 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×C8, C23×C4, C23×C4, C22.4Q16, C22×C4⋊C4, C23×C8, C2×C22.4Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×D8, C2×SD16, C2×Q16, C22.4Q16, C2×C2.C42, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C2×C2.D8, C2×C22.4Q16

Smallest permutation representation of C2×C22.4Q16
Regular action on 128 points
Generators in S128
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 64)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 94)(34 95)(35 96)(36 89)(37 90)(38 91)(39 92)(40 93)(49 77)(50 78)(51 79)(52 80)(53 73)(54 74)(55 75)(56 76)(65 88)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(97 116)(98 117)(99 118)(100 119)(101 120)(102 113)(103 114)(104 115)(105 124)(106 125)(107 126)(108 127)(109 128)(110 121)(111 122)(112 123)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(57 94)(58 95)(59 96)(60 89)(61 90)(62 91)(63 92)(64 93)(65 119)(66 120)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 108)(74 109)(75 110)(76 111)(77 112)(78 105)(79 106)(80 107)(81 101)(82 102)(83 103)(84 104)(85 97)(86 98)(87 99)(88 100)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 92)(18 93)(19 94)(20 95)(21 96)(22 89)(23 90)(24 91)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 104)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(65 108)(66 109)(67 110)(68 111)(69 112)(70 105)(71 106)(72 107)(73 119)(74 120)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 128)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)
(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 67 63 106)(2 120 64 78)(3 65 57 112)(4 118 58 76)(5 71 59 110)(6 116 60 74)(7 69 61 108)(8 114 62 80)(9 82 32 125)(10 101 25 50)(11 88 26 123)(12 99 27 56)(13 86 28 121)(14 97 29 54)(15 84 30 127)(16 103 31 52)(17 117 96 75)(18 70 89 109)(19 115 90 73)(20 68 91 107)(21 113 92 79)(22 66 93 105)(23 119 94 77)(24 72 95 111)(33 49 45 100)(34 122 46 87)(35 55 47 98)(36 128 48 85)(37 53 41 104)(38 126 42 83)(39 51 43 102)(40 124 44 81)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4H 4I ··· 4X 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C4 C4 D4 Q8 D4 D8 SD16 Q16 kernel C2×C22.4Q16 C22.4Q16 C22×C4⋊C4 C23×C8 C2×C4⋊C4 C22×C8 C22×C4 C22×C4 C24 C23 C23 C23 # reps 1 4 2 1 16 8 5 2 1 4 8 4

Matrix representation of C2×C22.4Q16 in GL6(𝔽17)

 1 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 1 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 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
,
 13 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 11 0 0 0 0 3 6
,
 4 0 0 0 0 0 0 16 0 0 0 0 0 0 10 16 0 0 0 0 16 7 0 0 0 0 0 0 7 7 0 0 0 0 5 10

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,11,6],[4,0,0,0,0,0,0,16,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,7,5,0,0,0,0,7,10] >;

C2×C22.4Q16 in GAP, Magma, Sage, TeX

C_2\times C_2^2._4Q_{16}
% in TeX

G:=Group("C2xC2^2.4Q16");
// GroupNames label

G:=SmallGroup(128,466);
// by ID

G=gap.SmallGroup(128,466);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,248,4037,124]);
// Polycyclic

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

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