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## G = (C2×C4).26D8order 128 = 27

### 19th non-split extension by C2×C4 of D8 acting via D8/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4).26D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.65C23 — (C2×C4).26D8
 Lower central C1 — C2 — C22×C4 — (C2×C4).26D8
 Upper central C1 — C23 — C2×C42 — (C2×C4).26D8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4).26D8

Generators and relations for (C2×C4).26D8
G = < a,b,c,d | a2=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=c-1 >

Subgroups: 224 in 112 conjugacy classes, 52 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22 [×3], C22 [×4], C8 [×3], C2×C4 [×2], C2×C4 [×6], C2×C4 [×19], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4⋊C8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16, C22.4Q16 [×2], C23.65C23 [×2], C2×C4⋊C8, C2×C2.D8, (C2×C4).26D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, D8 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×D8, C4○D8, C8.C22 [×2], C23.81C23, C87D4, C8.D4, D4⋊Q8 [×2], Q8.Q8 [×2], (C2×C4).26D8

Smallest permutation representation of (C2×C4).26D8
Regular action on 128 points
Generators in S128
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 117)(34 118)(35 119)(36 120)(37 113)(38 114)(39 115)(40 116)(49 93)(50 94)(51 95)(52 96)(53 89)(54 90)(55 91)(56 92)(73 108)(74 109)(75 110)(76 111)(77 112)(78 105)(79 106)(80 107)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)
(1 11 27 39)(2 40 28 12)(3 13 29 33)(4 34 30 14)(5 15 31 35)(6 36 32 16)(7 9 25 37)(8 38 26 10)(17 95 87 80)(18 73 88 96)(19 89 81 74)(20 75 82 90)(21 91 83 76)(22 77 84 92)(23 93 85 78)(24 79 86 94)(41 117 97 61)(42 62 98 118)(43 119 99 63)(44 64 100 120)(45 113 101 57)(46 58 102 114)(47 115 103 59)(48 60 104 116)(49 125 105 67)(50 68 106 126)(51 127 107 69)(52 70 108 128)(53 121 109 71)(54 72 110 122)(55 123 111 65)(56 66 112 124)
(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 94 27 79)(2 93 28 78)(3 92 29 77)(4 91 30 76)(5 90 31 75)(6 89 32 74)(7 96 25 73)(8 95 26 80)(9 128 37 70)(10 127 38 69)(11 126 39 68)(12 125 40 67)(13 124 33 66)(14 123 34 65)(15 122 35 72)(16 121 36 71)(17 58 87 114)(18 57 88 113)(19 64 81 120)(20 63 82 119)(21 62 83 118)(22 61 84 117)(23 60 85 116)(24 59 86 115)(41 112 97 56)(42 111 98 55)(43 110 99 54)(44 109 100 53)(45 108 101 52)(46 107 102 51)(47 106 103 50)(48 105 104 49)

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

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

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

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 2 2 2 2 2 2 4 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D4 D4 D8 C4○D4 C4○D8 C8.C22 kernel (C2×C4).26D8 C22.4Q16 C23.65C23 C2×C4⋊C8 C2×C2.D8 C4⋊C4 C2×C8 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 3 2 1 1 4 2 2 4 6 4 2

Matrix representation of (C2×C4).26D8 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
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 5 5 0 0 0 0 5 12 0 0 0 0 0 0 0 4 0 0 0 0 13 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 5 12 0 0 0 0 12 12 0 0 0 0 0 0 3 3 0 0 0 0 14 3
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 5 5 0 0 0 0 5 12 0 0 0 0 0 0 11 4 0 0 0 0 4 6

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

(C2×C4).26D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{26}D_8
% in TeX

G:=Group("(C2xC4).26D8");
// GroupNames label

G:=SmallGroup(128,818);
// by ID

G=gap.SmallGroup(128,818);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,58,718,172]);
// 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,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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