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## G = (C2×C8).60D4order 128 = 27

### 28th 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).60D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C23.67C23 — (C2×C8).60D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).60D4
 Upper central C1 — C23 — C2×C42 — (C2×C8).60D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).60D4

Generators and relations for (C2×C8).60D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab4, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b4c3 >

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], C2.D8 [×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×C2.D8 [×2], (C2×C8).60D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, Q16 [×4], C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C2×Q16 [×2], C8⋊C22 [×2], C23.4Q8, C4.Q16 [×2], C23.48D4 [×2], C4⋊Q16, C83D4, (C2×C8).60D4

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

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

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

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

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 2 2 2 2 2 4 type + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 D4 D4 Q8 Q16 C4○D4 C8⋊C22 kernel (C2×C8).60D4 C22.7C42 C42⋊9C4 C23.67C23 C2×Q8⋊C4 C2×C2.D8 C2×C8 C22×C4 C2×Q8 C2×C4 C2×C4 C22 # reps 1 1 1 1 2 2 4 2 2 8 6 2

Matrix representation of (C2×C8).60D4 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
,
 7 1 0 0 0 0 1 10 0 0 0 0 0 0 6 11 0 0 0 0 3 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 7 0 0 0 0 7 1 0 0 0 0 0 0 11 6 0 0 0 0 14 0 0 0 0 0 0 0 4 0 0 0 0 0 13 13
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 4 0 0 0 0 0 4 13 0 0 0 0 0 0 13 9 0 0 0 0 4 4

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],[7,1,0,0,0,0,1,10,0,0,0,0,0,0,6,3,0,0,0,0,11,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,7,0,0,0,0,7,1,0,0,0,0,0,0,11,14,0,0,0,0,6,0,0,0,0,0,0,0,4,13,0,0,0,0,0,13],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,0,13,0,0,0,0,0,0,13,4,0,0,0,0,9,4] >;

(C2×C8).60D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{60}D_4
% in TeX

G:=Group("(C2xC8).60D4");
// GroupNames label

G:=SmallGroup(128,827);
// by ID

G=gap.SmallGroup(128,827);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,352,1018,521,248,1411,718,172]);
// 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^-1,d*c*d^-1=b^4*c^3>;
// generators/relations

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