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## G = (C2×C8).52D4order 128 = 27

### 20th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).52D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×Q8⋊C4 — (C2×C8).52D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).52D4
 Upper central C1 — C23 — C2×C42 — (C2×C8).52D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).52D4

Generators and relations for (C2×C8).52D4
G = < a,b,c,d | a2=b8=c4=1, d2=b2, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=b6c-1 >

Subgroups: 248 in 124 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×19], Q8 [×6], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], Q8⋊C4 [×4], C4⋊C8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C2.D8, (C2×C8).52D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, Q16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, C23.Q8, C42Q16, D4.2D4, C8.18D4, C8⋊D4, C4.Q16, Q8.Q8, (C2×C8).52D4

Smallest permutation representation of (C2×C8).52D4
Regular action on 128 points
Generators in S128
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 127)(34 128)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 112)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(81 93)(82 94)(83 95)(84 96)(85 89)(86 90)(87 91)(88 92)
(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)
(1 121 23 94)(2 128 24 93)(3 127 17 92)(4 126 18 91)(5 125 19 90)(6 124 20 89)(7 123 21 96)(8 122 22 95)(9 58 106 102)(10 57 107 101)(11 64 108 100)(12 63 109 99)(13 62 110 98)(14 61 111 97)(15 60 112 104)(16 59 105 103)(25 48 65 114)(26 47 66 113)(27 46 67 120)(28 45 68 119)(29 44 69 118)(30 43 70 117)(31 42 71 116)(32 41 72 115)(33 80 88 55)(34 79 81 54)(35 78 82 53)(36 77 83 52)(37 76 84 51)(38 75 85 50)(39 74 86 49)(40 73 87 56)
(1 14 3 16 5 10 7 12)(2 115 4 117 6 119 8 113)(9 50 11 52 13 54 15 56)(17 105 19 107 21 109 23 111)(18 43 20 45 22 47 24 41)(25 86 27 88 29 82 31 84)(26 91 28 93 30 95 32 89)(33 69 35 71 37 65 39 67)(34 102 36 104 38 98 40 100)(42 74 44 76 46 78 48 80)(49 118 51 120 53 114 55 116)(57 94 59 96 61 90 63 92)(58 83 60 85 62 87 64 81)(66 126 68 128 70 122 72 124)(73 106 75 108 77 110 79 112)(97 125 99 127 101 121 103 123)

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

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

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

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 2 2 2 2 4 4 type + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 Q8 Q16 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel (C2×C8).52D4 C22.4Q16 C23.65C23 C23.67C23 C2×Q8⋊C4 C2×C4⋊C8 C2×C2.D8 C4⋊C4 C2×C8 C22×C4 C2×Q8 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 1 1 2 1 1 2 2 2 2 4 6 4 1 1

Matrix representation of (C2×C8).52D4 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 16 0 0 0 0 0 0 16
,
 3 3 0 0 0 0 14 3 0 0 0 0 0 0 6 2 0 0 0 0 7 11 0 0 0 0 0 0 13 9 0 0 0 0 0 4
,
 14 14 0 0 0 0 14 3 0 0 0 0 0 0 10 9 0 0 0 0 2 7 0 0 0 0 0 0 16 0 0 0 0 0 1 1
,
 12 5 0 0 0 0 12 12 0 0 0 0 0 0 7 8 0 0 0 0 15 10 0 0 0 0 0 0 16 15 0 0 0 0 1 1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,0,0,0,0,13,0,0,0,0,0,9,4],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,10,2,0,0,0,0,9,7,0,0,0,0,0,0,16,1,0,0,0,0,0,1],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,7,15,0,0,0,0,8,10,0,0,0,0,0,0,16,1,0,0,0,0,15,1] >;

(C2×C8).52D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{52}D_4
% in TeX

G:=Group("(C2xC8).52D4");
// GroupNames label

G:=SmallGroup(128,800);
// by ID

G=gap.SmallGroup(128,800);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,512,422,387,352,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^2,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=b^6*c^-1>;
// generators/relations

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