p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.24Q8, C8⋊C4⋊16C4, C42⋊9C4.5C2, C42.142(C2×C4), C42⋊8C4.7C2, (C22×C4).279D4, C23.752(C2×D4), C4.3(C42.C2), C2.9(C42⋊8C4), C4.59(C42⋊C2), C22.65(C8⋊C22), C22.4Q16.47C2, (C2×C42).255C22, (C22×C8).385C22, C2.8(M4(2)⋊C4), (C22×C4).1340C23, C22.57(C4.4D4), C22.54(C8.C22), C2.1(C42.28C22), C2.1(C42.30C22), C2.1(C42.29C22), (C2×C4).45(C4⋊C4), (C2×C8).144(C2×C4), C22.98(C2×C4⋊C4), (C2×C4).192(C2×Q8), (C2×C8⋊C4).28C2, (C2×C4⋊C4).46C22, (C2×C4).558(C4○D4), (C2×C4).538(C22×C4), SmallGroup(128,568)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.24Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2c3 >
Subgroups: 220 in 112 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2.C42, C8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C42⋊8C4, C42⋊9C4, C2×C8⋊C4, C42.24Q8
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⋊C22, C8.C22, C42⋊8C4, M4(2)⋊C4, C42.28C22, C42.29C22, C42.30C22, C42.24Q8
►Regular action on 128 points
Generators in S128
(1 79 21 62)(2 76 22 59)(3 73 23 64)(4 78 24 61)(5 75 17 58)(6 80 18 63)(7 77 19 60)(8 74 20 57)(9 54 46 32)(10 51 47 29)(11 56 48 26)(12 53 41 31)(13 50 42 28)(14 55 43 25)(15 52 44 30)(16 49 45 27)(33 95 125 71)(34 92 126 68)(35 89 127 65)(36 94 128 70)(37 91 121 67)(38 96 122 72)(39 93 123 69)(40 90 124 66)(81 107 115 101)(82 112 116 98)(83 109 117 103)(84 106 118 100)(85 111 119 97)(86 108 120 102)(87 105 113 99)(88 110 114 104)
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 59 13 63)(10 60 14 64)(11 61 15 57)(12 62 16 58)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)(33 104 37 100)(34 97 38 101)(35 98 39 102)(36 99 40 103)(41 79 45 75)(42 80 46 76)(43 73 47 77)(44 74 48 78)(65 116 69 120)(66 117 70 113)(67 118 71 114)(68 119 72 115)(81 92 85 96)(82 93 86 89)(83 94 87 90)(84 95 88 91)(105 124 109 128)(106 125 110 121)(107 126 111 122)(108 127 112 123)
(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 83 55 92)(2 120 56 71)(3 81 49 90)(4 118 50 69)(5 87 51 96)(6 116 52 67)(7 85 53 94)(8 114 54 65)(9 39 57 100)(10 126 58 109)(11 37 59 98)(12 124 60 107)(13 35 61 104)(14 122 62 105)(15 33 63 102)(16 128 64 111)(17 113 29 72)(18 82 30 91)(19 119 31 70)(20 88 32 89)(21 117 25 68)(22 86 26 95)(23 115 27 66)(24 84 28 93)(34 75 103 47)(36 73 97 45)(38 79 99 43)(40 77 101 41)(42 127 78 110)(44 125 80 108)(46 123 74 106)(48 121 76 112)
G:=sub<Sym(128)| (1,79,21,62)(2,76,22,59)(3,73,23,64)(4,78,24,61)(5,75,17,58)(6,80,18,63)(7,77,19,60)(8,74,20,57)(9,54,46,32)(10,51,47,29)(11,56,48,26)(12,53,41,31)(13,50,42,28)(14,55,43,25)(15,52,44,30)(16,49,45,27)(33,95,125,71)(34,92,126,68)(35,89,127,65)(36,94,128,70)(37,91,121,67)(38,96,122,72)(39,93,123,69)(40,90,124,66)(81,107,115,101)(82,112,116,98)(83,109,117,103)(84,106,118,100)(85,111,119,97)(86,108,120,102)(87,105,113,99)(88,110,114,104), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,59,13,63)(10,60,14,64)(11,61,15,57)(12,62,16,58)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,104,37,100)(34,97,38,101)(35,98,39,102)(36,99,40,103)(41,79,45,75)(42,80,46,76)(43,73,47,77)(44,74,48,78)(65,116,69,120)(66,117,70,113)(67,118,71,114)(68,119,72,115)(81,92,85,96)(82,93,86,89)(83,94,87,90)(84,95,88,91)(105,124,109,128)(106,125,110,121)(107,126,111,122)(108,127,112,123), (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), (1,83,55,92)(2,120,56,71)(3,81,49,90)(4,118,50,69)(5,87,51,96)(6,116,52,67)(7,85,53,94)(8,114,54,65)(9,39,57,100)(10,126,58,109)(11,37,59,98)(12,124,60,107)(13,35,61,104)(14,122,62,105)(15,33,63,102)(16,128,64,111)(17,113,29,72)(18,82,30,91)(19,119,31,70)(20,88,32,89)(21,117,25,68)(22,86,26,95)(23,115,27,66)(24,84,28,93)(34,75,103,47)(36,73,97,45)(38,79,99,43)(40,77,101,41)(42,127,78,110)(44,125,80,108)(46,123,74,106)(48,121,76,112)>;
G:=Group( (1,79,21,62)(2,76,22,59)(3,73,23,64)(4,78,24,61)(5,75,17,58)(6,80,18,63)(7,77,19,60)(8,74,20,57)(9,54,46,32)(10,51,47,29)(11,56,48,26)(12,53,41,31)(13,50,42,28)(14,55,43,25)(15,52,44,30)(16,49,45,27)(33,95,125,71)(34,92,126,68)(35,89,127,65)(36,94,128,70)(37,91,121,67)(38,96,122,72)(39,93,123,69)(40,90,124,66)(81,107,115,101)(82,112,116,98)(83,109,117,103)(84,106,118,100)(85,111,119,97)(86,108,120,102)(87,105,113,99)(88,110,114,104), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,59,13,63)(10,60,14,64)(11,61,15,57)(12,62,16,58)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,104,37,100)(34,97,38,101)(35,98,39,102)(36,99,40,103)(41,79,45,75)(42,80,46,76)(43,73,47,77)(44,74,48,78)(65,116,69,120)(66,117,70,113)(67,118,71,114)(68,119,72,115)(81,92,85,96)(82,93,86,89)(83,94,87,90)(84,95,88,91)(105,124,109,128)(106,125,110,121)(107,126,111,122)(108,127,112,123), (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), (1,83,55,92)(2,120,56,71)(3,81,49,90)(4,118,50,69)(5,87,51,96)(6,116,52,67)(7,85,53,94)(8,114,54,65)(9,39,57,100)(10,126,58,109)(11,37,59,98)(12,124,60,107)(13,35,61,104)(14,122,62,105)(15,33,63,102)(16,128,64,111)(17,113,29,72)(18,82,30,91)(19,119,31,70)(20,88,32,89)(21,117,25,68)(22,86,26,95)(23,115,27,66)(24,84,28,93)(34,75,103,47)(36,73,97,45)(38,79,99,43)(40,77,101,41)(42,127,78,110)(44,125,80,108)(46,123,74,106)(48,121,76,112) );
G=PermutationGroup([[(1,79,21,62),(2,76,22,59),(3,73,23,64),(4,78,24,61),(5,75,17,58),(6,80,18,63),(7,77,19,60),(8,74,20,57),(9,54,46,32),(10,51,47,29),(11,56,48,26),(12,53,41,31),(13,50,42,28),(14,55,43,25),(15,52,44,30),(16,49,45,27),(33,95,125,71),(34,92,126,68),(35,89,127,65),(36,94,128,70),(37,91,121,67),(38,96,122,72),(39,93,123,69),(40,90,124,66),(81,107,115,101),(82,112,116,98),(83,109,117,103),(84,106,118,100),(85,111,119,97),(86,108,120,102),(87,105,113,99),(88,110,114,104)], [(1,49,5,53),(2,50,6,54),(3,51,7,55),(4,52,8,56),(9,59,13,63),(10,60,14,64),(11,61,15,57),(12,62,16,58),(17,31,21,27),(18,32,22,28),(19,25,23,29),(20,26,24,30),(33,104,37,100),(34,97,38,101),(35,98,39,102),(36,99,40,103),(41,79,45,75),(42,80,46,76),(43,73,47,77),(44,74,48,78),(65,116,69,120),(66,117,70,113),(67,118,71,114),(68,119,72,115),(81,92,85,96),(82,93,86,89),(83,94,87,90),(84,95,88,91),(105,124,109,128),(106,125,110,121),(107,126,111,122),(108,127,112,123)], [(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)], [(1,83,55,92),(2,120,56,71),(3,81,49,90),(4,118,50,69),(5,87,51,96),(6,116,52,67),(7,85,53,94),(8,114,54,65),(9,39,57,100),(10,126,58,109),(11,37,59,98),(12,124,60,107),(13,35,61,104),(14,122,62,105),(15,33,63,102),(16,128,64,111),(17,113,29,72),(18,82,30,91),(19,119,31,70),(20,88,32,89),(21,117,25,68),(22,86,26,95),(23,115,27,66),(24,84,28,93),(34,75,103,47),(36,73,97,45),(38,79,99,43),(40,77,101,41),(42,127,78,110),(44,125,80,108),(46,123,74,106),(48,121,76,112)]])
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 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.24Q8 | C22.4Q16 | C42⋊8C4 | C42⋊9C4 | C2×C8⋊C4 | C8⋊C4 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.24Q8 ►in GL8(𝔽17)
| 1 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 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 | 13 |
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(17))| [1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,15,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0],[10,1,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,8,6,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.24Q8 in GAP, Magma, Sage, TeX
C_4^2._{24}Q_8 % in TeX
G:=Group("C4^2.24Q8"); // GroupNames label
G:=SmallGroup(128,568);
// by ID
G=gap.SmallGroup(128,568);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,723,58,2019,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^3>;
// generators/relations