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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

Aliases: (C2×C8).171D4, (C2×Q8).12Q8, (C22×C4).165D4, C23.944(C2×D4), C2.14(Q8.Q8), C4.44(C22⋊Q8), C428C4.17C2, C2.18(C8.2D4), C2.19(C8.12D4), C22.130(C4○D8), C22.83(C41D4), (C2×C42).395C22, (C22×C8).333C22, (C22×Q8).83C22, (C22×C4).1478C23, C4.38(C22.D4), C22.121(C22⋊Q8), C2.10(C23.4Q8), C2.14(C23.20D4), C22.149(C8.C22), C22.7C42.29C2, C23.67C23.23C2, C22.131(C22.D4), (C2×C4).753(C2×D4), (C2×C4).295(C2×Q8), (C2×C4.Q8).30C2, (C2×C2.D8).21C2, (C2×C4).784(C4○D4), (C2×C4⋊C4).167C22, (C2×Q8⋊C4).22C2, SmallGroup(128,829)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).171D4
C1C2C22C2×C4C22×C4C22×Q8C23.67C23 — (C2×C8).171D4
C1C2C22×C4 — (C2×C8).171D4
C1C23C2×C42 — (C2×C8).171D4
C1C2C2C22×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···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim1111111222224
type+++++++++--
imageC1C2C2C2C2C2C2D4D4Q8C4○D4C4○D8C8.C22
kernel(C2×C8).171D4C22.7C42C428C4C23.67C23C2×Q8⋊C4C2×C4.Q8C2×C2.D8C2×C8C22×C4C2×Q8C2×C4C22C22
# reps1111211422682

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

1600000
0160000
001000
000100
000010
000001
,
470000
10130000
0010400
0013700
000007
000057
,
610000
16110000
001000
000100
0000116
0000140
,
1300000
0130000
0041000
0071300
0000116
0000146

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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