p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.23Q8, C42.103D4, C42⋊4C4.7C2, C2.5(C42⋊8C4), C4.81(C4.4D4), C22.48(C8○D4), (C22×C8).20C22, C4.35(C42.C2), C2.C42.17C4, C23.309(C22×C4), (C2×C42).254C22, (C22×C4).1622C23, C22.7C42.4C2, C22.80(C42⋊C2), C2.8(C42.7C22), C2.11(C42.6C22), (C2×C4⋊C8).25C2, (C2×C4).44(C4⋊C4), C22.95(C2×C4⋊C4), (C2×C4).338(C2×Q8), (C2×C4).1520(C2×D4), (C2×C4).930(C4○D4), (C22×C4).114(C2×C4), SmallGroup(128,564)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.23Q8
G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2c-1 >
Subgroups: 172 in 108 conjugacy classes, 60 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C22×C4, C2.C42, C4⋊C8, C2×C42, C2×C42, C22×C8, C22.7C42, C42⋊4C4, C2×C4⋊C8, C42.23Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C8○D4, C42⋊8C4, C42.6C22, C42.7C22, C42.23Q8
►Regular action on 128 points
Generators in S128
(1 65 47 77)(2 78 48 66)(3 67 41 79)(4 80 42 68)(5 69 43 73)(6 74 44 70)(7 71 45 75)(8 76 46 72)(9 100 122 20)(10 21 123 101)(11 102 124 22)(12 23 125 103)(13 104 126 24)(14 17 127 97)(15 98 128 18)(16 19 121 99)(25 63 105 95)(26 96 106 64)(27 57 107 89)(28 90 108 58)(29 59 109 91)(30 92 110 60)(31 61 111 93)(32 94 112 62)(33 83 113 51)(34 52 114 84)(35 85 115 53)(36 54 116 86)(37 87 117 55)(38 56 118 88)(39 81 119 49)(40 50 120 82)
(1 97 5 101)(2 98 6 102)(3 99 7 103)(4 100 8 104)(9 72 13 68)(10 65 14 69)(11 66 15 70)(12 67 16 71)(17 43 21 47)(18 44 22 48)(19 45 23 41)(20 46 24 42)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)(73 123 77 127)(74 124 78 128)(75 125 79 121)(76 126 80 122)(81 107 85 111)(82 108 86 112)(83 109 87 105)(84 110 88 106)(89 115 93 119)(90 116 94 120)(91 117 95 113)(92 118 96 114)
(1 119 103 95)(2 64 104 40)(3 113 97 89)(4 58 98 34)(5 115 99 91)(6 60 100 36)(7 117 101 93)(8 62 102 38)(9 82 70 106)(10 27 71 51)(11 84 72 108)(12 29 65 53)(13 86 66 110)(14 31 67 55)(15 88 68 112)(16 25 69 49)(17 57 41 33)(18 114 42 90)(19 59 43 35)(20 116 44 92)(21 61 45 37)(22 118 46 94)(23 63 47 39)(24 120 48 96)(26 122 50 74)(28 124 52 76)(30 126 54 78)(32 128 56 80)(73 81 121 105)(75 83 123 107)(77 85 125 109)(79 87 127 111)
(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)
G:=sub<Sym(128)| (1,65,47,77)(2,78,48,66)(3,67,41,79)(4,80,42,68)(5,69,43,73)(6,74,44,70)(7,71,45,75)(8,76,46,72)(9,100,122,20)(10,21,123,101)(11,102,124,22)(12,23,125,103)(13,104,126,24)(14,17,127,97)(15,98,128,18)(16,19,121,99)(25,63,105,95)(26,96,106,64)(27,57,107,89)(28,90,108,58)(29,59,109,91)(30,92,110,60)(31,61,111,93)(32,94,112,62)(33,83,113,51)(34,52,114,84)(35,85,115,53)(36,54,116,86)(37,87,117,55)(38,56,118,88)(39,81,119,49)(40,50,120,82), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,72,13,68)(10,65,14,69)(11,66,15,70)(12,67,16,71)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,115,93,119)(90,116,94,120)(91,117,95,113)(92,118,96,114), (1,119,103,95)(2,64,104,40)(3,113,97,89)(4,58,98,34)(5,115,99,91)(6,60,100,36)(7,117,101,93)(8,62,102,38)(9,82,70,106)(10,27,71,51)(11,84,72,108)(12,29,65,53)(13,86,66,110)(14,31,67,55)(15,88,68,112)(16,25,69,49)(17,57,41,33)(18,114,42,90)(19,59,43,35)(20,116,44,92)(21,61,45,37)(22,118,46,94)(23,63,47,39)(24,120,48,96)(26,122,50,74)(28,124,52,76)(30,126,54,78)(32,128,56,80)(73,81,121,105)(75,83,123,107)(77,85,125,109)(79,87,127,111), (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)>;
G:=Group( (1,65,47,77)(2,78,48,66)(3,67,41,79)(4,80,42,68)(5,69,43,73)(6,74,44,70)(7,71,45,75)(8,76,46,72)(9,100,122,20)(10,21,123,101)(11,102,124,22)(12,23,125,103)(13,104,126,24)(14,17,127,97)(15,98,128,18)(16,19,121,99)(25,63,105,95)(26,96,106,64)(27,57,107,89)(28,90,108,58)(29,59,109,91)(30,92,110,60)(31,61,111,93)(32,94,112,62)(33,83,113,51)(34,52,114,84)(35,85,115,53)(36,54,116,86)(37,87,117,55)(38,56,118,88)(39,81,119,49)(40,50,120,82), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,72,13,68)(10,65,14,69)(11,66,15,70)(12,67,16,71)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,115,93,119)(90,116,94,120)(91,117,95,113)(92,118,96,114), (1,119,103,95)(2,64,104,40)(3,113,97,89)(4,58,98,34)(5,115,99,91)(6,60,100,36)(7,117,101,93)(8,62,102,38)(9,82,70,106)(10,27,71,51)(11,84,72,108)(12,29,65,53)(13,86,66,110)(14,31,67,55)(15,88,68,112)(16,25,69,49)(17,57,41,33)(18,114,42,90)(19,59,43,35)(20,116,44,92)(21,61,45,37)(22,118,46,94)(23,63,47,39)(24,120,48,96)(26,122,50,74)(28,124,52,76)(30,126,54,78)(32,128,56,80)(73,81,121,105)(75,83,123,107)(77,85,125,109)(79,87,127,111), (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) );
G=PermutationGroup([[(1,65,47,77),(2,78,48,66),(3,67,41,79),(4,80,42,68),(5,69,43,73),(6,74,44,70),(7,71,45,75),(8,76,46,72),(9,100,122,20),(10,21,123,101),(11,102,124,22),(12,23,125,103),(13,104,126,24),(14,17,127,97),(15,98,128,18),(16,19,121,99),(25,63,105,95),(26,96,106,64),(27,57,107,89),(28,90,108,58),(29,59,109,91),(30,92,110,60),(31,61,111,93),(32,94,112,62),(33,83,113,51),(34,52,114,84),(35,85,115,53),(36,54,116,86),(37,87,117,55),(38,56,118,88),(39,81,119,49),(40,50,120,82)], [(1,97,5,101),(2,98,6,102),(3,99,7,103),(4,100,8,104),(9,72,13,68),(10,65,14,69),(11,66,15,70),(12,67,16,71),(17,43,21,47),(18,44,22,48),(19,45,23,41),(20,46,24,42),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,59,37,63),(34,60,38,64),(35,61,39,57),(36,62,40,58),(73,123,77,127),(74,124,78,128),(75,125,79,121),(76,126,80,122),(81,107,85,111),(82,108,86,112),(83,109,87,105),(84,110,88,106),(89,115,93,119),(90,116,94,120),(91,117,95,113),(92,118,96,114)], [(1,119,103,95),(2,64,104,40),(3,113,97,89),(4,58,98,34),(5,115,99,91),(6,60,100,36),(7,117,101,93),(8,62,102,38),(9,82,70,106),(10,27,71,51),(11,84,72,108),(12,29,65,53),(13,86,66,110),(14,31,67,55),(15,88,68,112),(16,25,69,49),(17,57,41,33),(18,114,42,90),(19,59,43,35),(20,116,44,92),(21,61,45,37),(22,118,46,94),(23,63,47,39),(24,120,48,96),(26,122,50,74),(28,124,52,76),(30,126,54,78),(32,128,56,80),(73,81,121,105),(75,83,123,107),(77,85,125,109),(79,87,127,111)], [(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)]])
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | C4○D4 | C8○D4 |
| kernel | C42.23Q8 | C22.7C42 | C42⋊4C4 | C2×C4⋊C8 | C2.C42 | C42 | C42 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 8 | 2 | 2 | 8 | 16 |
Matrix representation of C42.23Q8 ►in GL6(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 6 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 13 |
| 0 | 0 | 0 | 0 | 4 | 10 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 14 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 8 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 8 |
| 0 | 0 | 0 | 0 | 9 | 14 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,13,9,0,0,0,0,0,4,0,0,0,0,0,0,1,6,0,0,0,0,11,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,7,4,0,0,0,0,13,10],[5,14,0,0,0,0,3,12,0,0,0,0,0,0,4,8,0,0,0,0,13,13,0,0,0,0,0,0,3,9,0,0,0,0,8,14] >;
C42.23Q8 in GAP, Magma, Sage, TeX
C_4^2._{23}Q_8 % in TeX
G:=Group("C4^2.23Q8"); // GroupNames label
G:=SmallGroup(128,564);
// by ID
G=gap.SmallGroup(128,564);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,723,58,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b*c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations