Copied to
clipboard

## G = C42.120D4order 128 = 27

### 102nd non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.120D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.120D4
 Lower central C1 — C23 — C42.120D4
 Upper central C1 — C22×C4 — C42.120D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.120D4

Generators and relations for C42.120D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 220 in 140 conjugacy classes, 68 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×6], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×10], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C8⋊C4 [×2], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×4], C22×C8 [×4], C22×Q8, C22.7C42 [×4], C2×C8⋊C4, C2×C4⋊C8, C2×C4×Q8, C42.120D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C2×M4(2) [×2], C8○D4 [×2], C23.67C23, C24.4C4, (C22×C8)⋊C2, C84Q8 [×4], C42.120D4

Smallest permutation representation of C42.120D4
Regular action on 128 points
Generators in S128
(1 80 27 71)(2 77 28 68)(3 74 29 65)(4 79 30 70)(5 76 31 67)(6 73 32 72)(7 78 25 69)(8 75 26 66)(9 82 109 126)(10 87 110 123)(11 84 111 128)(12 81 112 125)(13 86 105 122)(14 83 106 127)(15 88 107 124)(16 85 108 121)(17 104 63 54)(18 101 64 51)(19 98 57 56)(20 103 58 53)(21 100 59 50)(22 97 60 55)(23 102 61 52)(24 99 62 49)(33 44 117 90)(34 41 118 95)(35 46 119 92)(36 43 120 89)(37 48 113 94)(38 45 114 91)(39 42 115 96)(40 47 116 93)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 90 13 94)(10 91 14 95)(11 92 15 96)(12 93 16 89)(25 63 29 59)(26 64 30 60)(27 57 31 61)(28 58 32 62)(33 86 37 82)(34 87 38 83)(35 88 39 84)(36 81 40 85)(41 110 45 106)(42 111 46 107)(43 112 47 108)(44 105 48 109)(49 68 53 72)(50 69 54 65)(51 70 55 66)(52 71 56 67)(73 99 77 103)(74 100 78 104)(75 101 79 97)(76 102 80 98)(113 126 117 122)(114 127 118 123)(115 128 119 124)(116 121 120 125)
(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 86 31 126)(2 120 32 40)(3 88 25 128)(4 114 26 34)(5 82 27 122)(6 116 28 36)(7 84 29 124)(8 118 30 38)(9 67 105 80)(10 101 106 55)(11 69 107 74)(12 103 108 49)(13 71 109 76)(14 97 110 51)(15 65 111 78)(16 99 112 53)(17 35 59 115)(18 123 60 83)(19 37 61 117)(20 125 62 85)(21 39 63 119)(22 127 64 87)(23 33 57 113)(24 121 58 81)(41 66 91 79)(42 100 92 54)(43 68 93 73)(44 102 94 56)(45 70 95 75)(46 104 96 50)(47 72 89 77)(48 98 90 52)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 M4(2) C4○D4 C8○D4 kernel C42.120D4 C22.7C42 C2×C8⋊C4 C2×C4⋊C8 C2×C4×Q8 C2×C4⋊C4 C22×Q8 C42 C2×C8 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 6 2 4 4 8 4 8

Matrix representation of C42.120D4 in GL6(𝔽17)

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

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

C42.120D4 in GAP, Magma, Sage, TeX

C_4^2._{120}D_4
% in TeX

G:=Group("C4^2.120D4");
// GroupNames label

G:=SmallGroup(128,717);
// by ID

G=gap.SmallGroup(128,717);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,723,100,124]);
// Polycyclic

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

׿
×
𝔽