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G = (C2×C8).170D4order 128 = 27

138th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).170D4, (C2×Q8).11Q8, (C2×C4).44SD16, C2.15(C85D4), (C22×C4).327D4, C23.943(C2×D4), C4.43(C22⋊Q8), C429C4.19C2, C2.10(Q8⋊Q8), C2.17(C8.2D4), C22.82(C41D4), (C2×C42).394C22, (C22×C8).332C22, C22.109(C2×SD16), C2.9(C23.4Q8), (C22×Q8).82C22, (C22×C4).1477C23, C4.37(C22.D4), C22.120(C22⋊Q8), C2.10(C23.47D4), C22.148(C8.C22), C22.7C42.37C2, C23.67C23.22C2, C22.130(C22.D4), (C2×C4).752(C2×D4), (C2×C4).294(C2×Q8), (C2×C4.Q8).29C2, (C2×C4).783(C4○D4), (C2×C4⋊C4).166C22, (C2×Q8⋊C4).28C2, SmallGroup(128,828)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).170D4
C1C2C22C2×C4C22×C4C22×Q8C23.67C23 — (C2×C8).170D4
C1C2C22×C4 — (C2×C8).170D4
C1C23C2×C42 — (C2×C8).170D4
C1C2C2C22×C4 — (C2×C8).170D4

Generators and relations for (C2×C8).170D4
 G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab4, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b3, dcd-1=c3 >

Subgroups: 256 in 127 conjugacy classes, 54 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], Q8 [×6], C23, C42 [×2], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×2], Q8⋊C4 [×4], C4.Q8 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×Q8, C22.7C42, C429C4, C23.67C23, C2×Q8⋊C4 [×2], C2×C4.Q8 [×2], (C2×C8).170D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×4], C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C2×SD16 [×2], C8.C22 [×2], C23.4Q8, Q8⋊Q8 [×2], C23.47D4 [×2], C85D4, C8.2D4, (C2×C8).170D4

Smallest permutation representation of (C2×C8).170D4
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 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 76)(18 77)(19 78)(20 79)(21 80)(22 73)(23 74)(24 75)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(33 100)(34 101)(35 102)(36 103)(37 104)(38 97)(39 98)(40 99)(41 96)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 112)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(65 120)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)
(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 37 65 47 5 33 69 43)(2 97 66 95 6 101 70 91)(3 39 67 41 7 35 71 45)(4 99 68 89 8 103 72 93)(9 84 110 78 13 88 106 74)(10 25 111 20 14 29 107 24)(11 86 112 80 15 82 108 76)(12 27 105 22 16 31 109 18)(17 62 26 49 21 58 30 53)(19 64 28 51 23 60 32 55)(34 117 44 123 38 113 48 127)(36 119 46 125 40 115 42 121)(50 73 59 83 54 77 63 87)(52 75 61 85 56 79 57 81)(90 122 104 120 94 126 100 116)(92 124 98 114 96 128 102 118)
(1 78 126 23)(2 73 127 18)(3 76 128 21)(4 79 121 24)(5 74 122 19)(6 77 123 22)(7 80 124 17)(8 75 125 20)(9 94 64 43)(10 89 57 46)(11 92 58 41)(12 95 59 44)(13 90 60 47)(14 93 61 42)(15 96 62 45)(16 91 63 48)(25 72 81 115)(26 67 82 118)(27 70 83 113)(28 65 84 116)(29 68 85 119)(30 71 86 114)(31 66 87 117)(32 69 88 120)(33 110 104 51)(34 105 97 54)(35 108 98 49)(36 111 99 52)(37 106 100 55)(38 109 101 50)(39 112 102 53)(40 107 103 56)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim111111222224
type++++++++--
imageC1C2C2C2C2C2D4D4Q8SD16C4○D4C8.C22
kernel(C2×C8).170D4C22.7C42C429C4C23.67C23C2×Q8⋊C4C2×C4.Q8C2×C8C22×C4C2×Q8C2×C4C2×C4C22
# reps111122422862

Matrix representation of (C2×C8).170D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
3120000
5140000
005300
00141200
000077
000050
,
14120000
530000
0016000
0001600
000007
0000510
,
400000
040000
00141200
005300
0000130
000044

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],[3,5,0,0,0,0,12,14,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,7,5,0,0,0,0,7,0],[14,5,0,0,0,0,12,3,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,0,0,0,7,10],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,14,5,0,0,0,0,12,3,0,0,0,0,0,0,13,4,0,0,0,0,0,4] >;

(C2×C8).170D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{170}D_4
% in TeX

G:=Group("(C2xC8).170D4");
// GroupNames label

G:=SmallGroup(128,828);
// by ID

G=gap.SmallGroup(128,828);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,504,141,176,422,387,352,1018,521,248,1411,718,172]);
// Polycyclic

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

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