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## G = C4.(C4×Q8)  order 128 = 27

### 8th non-split extension by C4 of C4×Q8 acting via C4×Q8/C42=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.(C4×Q8)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C4.(C4×Q8)
 Lower central C1 — C2 — C2×C4 — C4.(C4×Q8)
 Upper central C1 — C23 — C2×C42 — C4.(C4×Q8)
 Jennings C1 — C2 — C2 — C22×C4 — C4.(C4×Q8)

Generators and relations for C4.(C4×Q8)
G = < a,b,c,d | a4=c4=1, b4=a2, d2=ac2, ab=ba, cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=ac-1 >

Subgroups: 228 in 120 conjugacy classes, 64 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×20], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×2], C23.65C23 [×2], C2×C8⋊C4, C2×C4.Q8, C2×C2.D8, C4.(C4×Q8)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C8⋊C22 [×2], C8.C22 [×2], C23.65C23, Q16⋊C4, D8⋊C4, C8⋊D4 [×2], C8⋊Q8 [×2], C4.(C4×Q8)

Smallest permutation representation of C4.(C4×Q8)
Regular action on 128 points
Generators in S128
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 33 13 37)(10 34 14 38)(11 35 15 39)(12 36 16 40)(17 94 21 90)(18 95 22 91)(19 96 23 92)(20 89 24 93)(25 43 29 47)(26 44 30 48)(27 45 31 41)(28 46 32 42)(49 84 53 88)(50 85 54 81)(51 86 55 82)(52 87 56 83)(65 80 69 76)(66 73 70 77)(67 74 71 78)(68 75 72 79)(97 122 101 126)(98 123 102 127)(99 124 103 128)(100 125 104 121)(105 114 109 118)(106 115 110 119)(107 116 111 120)(108 117 112 113)
(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)
(1 107 95 74)(2 113 96 68)(3 105 89 80)(4 119 90 66)(5 111 91 78)(6 117 92 72)(7 109 93 76)(8 115 94 70)(9 102 42 86)(10 124 43 52)(11 100 44 84)(12 122 45 50)(13 98 46 82)(14 128 47 56)(15 104 48 88)(16 126 41 54)(17 77 58 110)(18 71 59 116)(19 75 60 108)(20 69 61 114)(21 73 62 106)(22 67 63 120)(23 79 64 112)(24 65 57 118)(25 87 38 103)(26 53 39 125)(27 85 40 101)(28 51 33 123)(29 83 34 99)(30 49 35 121)(31 81 36 97)(32 55 37 127)
(1 43 22 34 5 47 18 38)(2 48 23 39 6 44 19 35)(3 45 24 36 7 41 20 40)(4 42 17 33 8 46 21 37)(9 58 28 94 13 62 32 90)(10 63 29 91 14 59 25 95)(11 60 30 96 15 64 26 92)(12 57 31 93 16 61 27 89)(49 108 100 72 53 112 104 68)(50 105 101 69 54 109 97 65)(51 110 102 66 55 106 98 70)(52 107 103 71 56 111 99 67)(73 82 115 123 77 86 119 127)(74 87 116 128 78 83 120 124)(75 84 117 125 79 88 113 121)(76 81 118 122 80 85 114 126)

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

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

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

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 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 D4 Q8 D4 C4○D4 C8⋊C22 C8.C22 kernel C4.(C4×Q8) C22.4Q16 C23.65C23 C2×C8⋊C4 C2×C4.Q8 C2×C2.D8 C2.D8 C2×C8 C2×C8 C22×C4 C2×C4 C22 C22 # reps 1 2 2 1 1 1 8 2 4 2 4 2 2

Matrix representation of C4.(C4×Q8) in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 16 16 15 0 0 0 0 1 0 1 1
,
 13 15 0 0 0 0 0 0 16 4 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 12 3 15 7 0 0 0 0 6 14 9 11 0 0 0 0 13 5 8 16 0 0 0 0 0 12 14 0
,
 13 0 0 0 0 0 0 0 16 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 4 5 7 11 0 0 0 0 1 3 13 3 0 0 0 0 0 10 15 14 0 0 0 0 9 9 16 12
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 1 1 1 2 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 16

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[13,16,0,0,0,0,0,0,15,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,12,6,13,0,0,0,0,0,3,14,5,12,0,0,0,0,15,9,8,14,0,0,0,0,7,11,16,0],[13,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,1,0,9,0,0,0,0,5,3,10,9,0,0,0,0,7,13,15,16,0,0,0,0,11,3,14,12],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,16,0,0,0,0,0,1,16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,2,0,16] >;

C4.(C4×Q8) in GAP, Magma, Sage, TeX

C_4.(C_4\times Q_8)
% in TeX

G:=Group("C4.(C4xQ8)");
// GroupNames label

G:=SmallGroup(128,675);
// by ID

G=gap.SmallGroup(128,675);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,100,2804,172]);
// Polycyclic

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

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