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## G = C4.SD32order 128 = 27

### 4th non-split extension by C4 of SD32 acting via SD32/C16=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C4.SD32
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C4×C16 — C4.SD32
 Lower central C1 — C2 — C4 — C2×C8 — C4.SD32
 Upper central C1 — C22 — C42 — C4×C8 — C4.SD32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C4.SD32

Generators and relations for C4.SD32
G = < a,b,c | a4=b16=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b7 >

Subgroups: 168 in 70 conjugacy classes, 36 normal (18 characteristic)
C1, C2 [×3], C4 [×2], C4 [×4], C4 [×4], C22, C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×6], C16 [×2], C42, C4⋊C4 [×5], C2×C8 [×2], Q16 [×6], C2×Q8 [×3], C4×C8, C2.D8 [×2], C2.D8, C2×C16 [×2], C4⋊Q8 [×2], C2×Q16 [×2], C2×Q16, C4×C16, C2.Q32 [×4], C4⋊Q16, C82Q8, C4.SD32
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], SD32 [×2], Q32 [×2], C4.4D4, C2×D8, C2×SD16, C4.4D8, C2×SD32, C2×Q32, C4.SD32

Smallest permutation representation of C4.SD32
Regular action on 128 points
Generators in S128
(1 117 61 111)(2 118 62 112)(3 119 63 97)(4 120 64 98)(5 121 49 99)(6 122 50 100)(7 123 51 101)(8 124 52 102)(9 125 53 103)(10 126 54 104)(11 127 55 105)(12 128 56 106)(13 113 57 107)(14 114 58 108)(15 115 59 109)(16 116 60 110)(17 76 87 45)(18 77 88 46)(19 78 89 47)(20 79 90 48)(21 80 91 33)(22 65 92 34)(23 66 93 35)(24 67 94 36)(25 68 95 37)(26 69 96 38)(27 70 81 39)(28 71 82 40)(29 72 83 41)(30 73 84 42)(31 74 85 43)(32 75 86 44)
(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 76 61 45)(2 36 62 67)(3 74 63 43)(4 34 64 65)(5 72 49 41)(6 48 50 79)(7 70 51 39)(8 46 52 77)(9 68 53 37)(10 44 54 75)(11 66 55 35)(12 42 56 73)(13 80 57 33)(14 40 58 71)(15 78 59 47)(16 38 60 69)(17 111 87 117)(18 124 88 102)(19 109 89 115)(20 122 90 100)(21 107 91 113)(22 120 92 98)(23 105 93 127)(24 118 94 112)(25 103 95 125)(26 116 96 110)(27 101 81 123)(28 114 82 108)(29 99 83 121)(30 128 84 106)(31 97 85 119)(32 126 86 104)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 8A ··· 8H 16A ··· 16P order 1 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 16 ··· 16 size 1 1 1 1 2 ··· 2 16 16 16 16 2 ··· 2 2 ··· 2

38 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 D4 D4 SD16 C4○D4 D8 SD32 Q32 kernel C4.SD32 C4×C16 C2.Q32 C4⋊Q16 C8⋊2Q8 C42 C2×C8 C8 C8 C2×C4 C4 C4 # reps 1 1 4 1 1 1 1 4 4 4 8 8

Matrix representation of C4.SD32 in GL4(𝔽17) generated by

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

C4.SD32 in GAP, Magma, Sage, TeX

C_4.{\rm SD}_{32}
% in TeX

G:=Group("C4.SD32");
// GroupNames label

G:=SmallGroup(128,973);
// by ID

G=gap.SmallGroup(128,973);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,120,422,58,1123,360,3924,102,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^16=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^7>;
// generators/relations

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