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## G = (C2×C8).171D4order 128 = 27

### 139th 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).171D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C23.67C23 — (C2×C8).171D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).171D4
 Upper central C1 — C23 — C2×C42 — (C2×C8).171D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).171D4

Generators and relations for (C2×C8).171D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=a, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b3, dcd-1=b4c3 >

Subgroups: 240 in 119 conjugacy classes, 50 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×19], Q8 [×6], C23, C42 [×2], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×3], C22×C8 [×2], C22×Q8, C22.7C42, C428C4, C23.67C23, C2×Q8⋊C4 [×2], C2×C4.Q8, C2×C2.D8, (C2×C8).171D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C4○D8 [×2], C8.C22 [×2], C23.4Q8, Q8.Q8 [×2], C23.20D4 [×2], C8.12D4, C8.2D4, (C2×C8).171D4

Smallest permutation representation of (C2×C8).171D4
Regular action on 128 points
Generators in S128
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 121)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 105)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 104)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)
(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 36 62 87 5 40 58 83)(2 22 63 95 6 18 59 91)(3 38 64 81 7 34 60 85)(4 24 57 89 8 20 61 93)(9 84 123 37 13 88 127 33)(10 92 124 23 14 96 128 19)(11 86 125 39 15 82 121 35)(12 94 126 17 16 90 122 21)(25 111 102 115 29 107 98 119)(26 55 103 68 30 51 99 72)(27 105 104 117 31 109 100 113)(28 49 97 70 32 53 101 66)(41 50 75 71 45 54 79 67)(42 108 76 120 46 112 80 116)(43 52 77 65 47 56 73 69)(44 110 78 114 48 106 74 118)
(1 78 122 101)(2 73 123 104)(3 76 124 99)(4 79 125 102)(5 74 126 97)(6 77 127 100)(7 80 128 103)(8 75 121 98)(9 31 59 43)(10 26 60 46)(11 29 61 41)(12 32 62 44)(13 27 63 47)(14 30 64 42)(15 25 57 45)(16 28 58 48)(17 49 40 106)(18 52 33 109)(19 55 34 112)(20 50 35 107)(21 53 36 110)(22 56 37 105)(23 51 38 108)(24 54 39 111)(65 88 113 95)(66 83 114 90)(67 86 115 93)(68 81 116 96)(69 84 117 91)(70 87 118 94)(71 82 119 89)(72 85 120 92)

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

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

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

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 4 type + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D4 D4 Q8 C4○D4 C4○D8 C8.C22 kernel (C2×C8).171D4 C22.7C42 C42⋊8C4 C23.67C23 C2×Q8⋊C4 C2×C4.Q8 C2×C2.D8 C2×C8 C22×C4 C2×Q8 C2×C4 C22 C22 # reps 1 1 1 1 2 1 1 4 2 2 6 8 2

Matrix representation of (C2×C8).171D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 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
,
 4 7 0 0 0 0 10 13 0 0 0 0 0 0 10 4 0 0 0 0 13 7 0 0 0 0 0 0 0 7 0 0 0 0 5 7
,
 6 1 0 0 0 0 16 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 6 0 0 0 0 14 0
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 10 0 0 0 0 7 13 0 0 0 0 0 0 11 6 0 0 0 0 14 6

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

(C2×C8).171D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{171}D_4
% in TeX

G:=Group("(C2xC8).171D4");
// GroupNames label

G:=SmallGroup(128,829);
// by ID

G=gap.SmallGroup(128,829);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,624,422,387,352,1018,521,248,1411,718,172]);
// Polycyclic

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

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