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## G = C42.40Q8order 128 = 27

### 40th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.40Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.81C23 — C42.40Q8
 Lower central C1 — C23 — C42.40Q8
 Upper central C1 — C23 — C42.40Q8
 Jennings C1 — C23 — C42.40Q8

Generators and relations for C42.40Q8
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 308 in 170 conjugacy classes, 92 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C428C4, C23.81C23, C42.40Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2+ 1+4, 2- 1+4, C23.41C23, C22.56C24, C22.58C24, C42.40Q8

Character table of C42.40Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R size 1 1 1 1 1 1 1 1 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ10 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ11 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ12 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ14 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ15 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ16 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ17 2 -2 2 -2 2 -2 2 -2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 2 -2 2 -2 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 -2 2 -2 2 -2 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 -4 -4 4 -4 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 -4 4 4 -4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ23 4 -4 -4 -4 4 4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ24 4 4 -4 4 4 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ25 4 4 4 -4 -4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ26 4 4 -4 -4 -4 -4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C42.40Q8
Regular action on 128 points
Generators in S128
```(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 76 103 13)(2 73 104 14)(3 74 101 15)(4 75 102 16)(5 36 70 95)(6 33 71 96)(7 34 72 93)(8 35 69 94)(9 107 40 48)(10 108 37 45)(11 105 38 46)(12 106 39 47)(17 56 80 115)(18 53 77 116)(19 54 78 113)(20 55 79 114)(21 111 84 52)(22 112 81 49)(23 109 82 50)(24 110 83 51)(25 121 88 62)(26 122 85 63)(27 123 86 64)(28 124 87 61)(29 58 92 117)(30 59 89 118)(31 60 90 119)(32 57 91 120)(41 126 97 68)(42 127 98 65)(43 128 99 66)(44 125 100 67)
(1 78 38 52)(2 18 39 110)(3 80 40 50)(4 20 37 112)(5 30 43 122)(6 92 44 62)(7 32 41 124)(8 90 42 64)(9 109 101 17)(10 49 102 79)(11 111 103 19)(12 51 104 77)(13 113 105 21)(14 53 106 83)(15 115 107 23)(16 55 108 81)(22 75 114 45)(24 73 116 47)(25 96 117 67)(26 36 118 128)(27 94 119 65)(28 34 120 126)(29 100 121 71)(31 98 123 69)(33 58 125 88)(35 60 127 86)(46 84 76 54)(48 82 74 56)(57 68 87 93)(59 66 85 95)(61 72 91 97)(63 70 89 99)
(1 27 11 60)(2 87 12 120)(3 25 9 58)(4 85 10 118)(5 53 99 24)(6 113 100 84)(7 55 97 22)(8 115 98 82)(13 62 46 29)(14 122 47 89)(15 64 48 31)(16 124 45 91)(17 67 50 33)(18 126 51 93)(19 65 52 35)(20 128 49 95)(21 71 54 44)(23 69 56 42)(26 37 59 102)(28 39 57 104)(30 73 63 106)(32 75 61 108)(34 77 68 110)(36 79 66 112)(38 119 103 86)(40 117 101 88)(41 81 72 114)(43 83 70 116)(74 123 107 90)(76 121 105 92)(78 127 111 94)(80 125 109 96)```

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

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

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

Matrix representation of C42.40Q8 in GL12(𝔽5)

 0 1 1 3 0 0 0 0 0 0 0 0 4 0 3 4 0 0 0 0 0 0 0 0 4 2 0 1 0 0 0 0 0 0 0 0 2 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 4 0 0 0 0 0 0 0 0 0 1 0 3 2 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 2 0 0 0 0 0 0 0 0 2 4 4 2 0 0 0 0 0 0 0 0 0 1 3 3 0 0 0 0 0 0 0 0 1 1 2 3
,
 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 0 0 3 3 0 1 0 0 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 4 4 0 0 0 0 0 0 0 0 0 0 4 4 1
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 2 0 4 0 0 0 0 0 0 0 0 3 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 4 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 4 1
,
 3 1 0 1 0 0 0 0 0 0 0 0 1 2 4 0 0 0 0 0 0 0 0 0 0 4 3 1 0 0 0 0 0 0 0 0 1 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 1 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0

`G:=sub<GL(12,GF(5))| [0,4,4,2,0,0,0,0,0,0,0,0,1,0,2,1,0,0,0,0,0,0,0,0,1,3,0,4,0,0,0,0,0,0,0,0,3,4,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,3,1,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,4,1,1,0,0,0,0,0,0,0,0,4,4,3,2,0,0,0,0,0,0,0,0,2,2,3,3],[0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,3,0,0,0,0,0,0,0,0,0,2,4,3,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,1,0,4,4,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,2,3,0,0,0,0,0,0,0,0,3,1,2,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,4,0,4,0,0,0,0,0,0,0,0,0,0,0,1],[3,1,0,1,0,0,0,0,0,0,0,0,1,2,4,0,0,0,0,0,0,0,0,0,0,4,3,1,0,0,0,0,0,0,0,0,1,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,2,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,1,4,1,0] >;`

C42.40Q8 in GAP, Magma, Sage, TeX

`C_4^2._{40}Q_8`
`% in TeX`

`G:=Group("C4^2.40Q8");`
`// GroupNames label`

`G:=SmallGroup(128,1577);`
`// by ID`

`G=gap.SmallGroup(128,1577);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,456,758,723,436,794,185,80]);`
`// Polycyclic`

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

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