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## G = C2×C4.SD16order 128 = 27

### Direct product of C2 and C4.SD16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4.SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C2×C4.SD16
 Lower central C1 — C2 — C2×C4 — C2×C4.SD16
 Upper central C1 — C23 — C2×C42 — C2×C4.SD16
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4.SD16

Generators and relations for C2×C4.SD16
G = < a,b,c,d | a2=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c3 >

Subgroups: 372 in 212 conjugacy classes, 116 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×20], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×18], C4×C8 [×4], Q8⋊C4 [×16], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C22×Q8, C2×C4×C8, C2×Q8⋊C4 [×4], C4.SD16 [×8], C2×C4⋊Q8 [×2], C2×C4.SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], Q16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C2×SD16 [×6], C2×Q16 [×6], C22×D4, C2×C4○D4 [×2], C4.SD16 [×4], C2×C4.4D4, C22×SD16, C22×Q16, C2×C4.SD16

Smallest permutation representation of C2×C4.SD16
Regular action on 128 points
Generators in S128
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 80)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)(97 128)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(111 113)(112 114)
(1 62 11 31)(2 63 12 32)(3 64 13 25)(4 57 14 26)(5 58 15 27)(6 59 16 28)(7 60 9 29)(8 61 10 30)(17 54 73 38)(18 55 74 39)(19 56 75 40)(20 49 76 33)(21 50 77 34)(22 51 78 35)(23 52 79 36)(24 53 80 37)(41 95 99 113)(42 96 100 114)(43 89 101 115)(44 90 102 116)(45 91 103 117)(46 92 104 118)(47 93 97 119)(48 94 98 120)(65 81 122 111)(66 82 123 112)(67 83 124 105)(68 84 125 106)(69 85 126 107)(70 86 127 108)(71 87 128 109)(72 88 121 110)
(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 96 11 114)(2 117 12 91)(3 94 13 120)(4 115 14 89)(5 92 15 118)(6 113 16 95)(7 90 9 116)(8 119 10 93)(17 124 73 67)(18 70 74 127)(19 122 75 65)(20 68 76 125)(21 128 77 71)(22 66 78 123)(23 126 79 69)(24 72 80 121)(25 98 64 48)(26 43 57 101)(27 104 58 46)(28 41 59 99)(29 102 60 44)(30 47 61 97)(31 100 62 42)(32 45 63 103)(33 84 49 106)(34 109 50 87)(35 82 51 112)(36 107 52 85)(37 88 53 110)(38 105 54 83)(39 86 55 108)(40 111 56 81)

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

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

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

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 2 type + + + + + + + - image C1 C2 C2 C2 C2 D4 D4 SD16 Q16 C4○D4 kernel C2×C4.SD16 C2×C4×C8 C2×Q8⋊C4 C4.SD16 C2×C4⋊Q8 C42 C22×C4 C2×C4 C2×C4 C2×C4 # reps 1 1 4 8 2 2 2 8 8 8

Matrix representation of C2×C4.SD16 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 13 0 0 0 0 13 0 0 0 0 0 0 0 13 9 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 5 5 0 0 0 0 12 5
,
 5 3 0 0 0 0 14 12 0 0 0 0 0 0 16 15 0 0 0 0 1 1 0 0 0 0 0 0 6 4 0 0 0 0 4 11

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],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,0,9,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[5,14,0,0,0,0,3,12,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,6,4,0,0,0,0,4,11] >;

C2×C4.SD16 in GAP, Magma, Sage, TeX

C_2\times C_4.{\rm SD}_{16}
% in TeX

G:=Group("C2xC4.SD16");
// GroupNames label

G:=SmallGroup(128,1861);
// by ID

G=gap.SmallGroup(128,1861);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,2804,172,4037,124]);
// Polycyclic

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

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