p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.6Q8, (C2×C4).23Q16, (C2×C4).46SD16, C23.946(C2×D4), (C22×C4).329D4, C22.65(C2×Q16), C42⋊9C4.21C2, C2.11(C4.Q16), C2.11(D4⋊2Q8), (C22×C8).91C22, C2.8(C4.SD16), C4.26(C42.C2), C4.23(C42⋊2C2), C22.4Q16.31C2, (C2×C42).397C22, C22.111(C2×SD16), C22.161(C8⋊C22), (C22×C4).1480C23, C22.98(C4.4D4), C22.123(C22⋊Q8), C2.11(C23.46D4), C2.11(C23.48D4), C22.7C42.16C2, C23.65C23.27C2, C2.8(C42.29C22), C2.6(C23.83C23), C22.133(C22.D4), (C2×C4).297(C2×Q8), (C2×C4).786(C4○D4), (C2×C4⋊C4).169C22, SmallGroup(128,832)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4).23Q16
G = < a,b,c,d | a2=b4=c8=1, d2=b2c4, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd-1=ac-1 >
Subgroups: 224 in 107 conjugacy classes, 50 normal (28 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C42⋊9C4, C23.65C23, (C2×C4).23Q16
Quotients: C1, C2, C22, D4, Q8, C23, SD16, Q16, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×SD16, C2×Q16, C8⋊C22, C23.83C23, D4⋊2Q8, C4.Q16, C23.46D4, C23.48D4, C4.SD16, C42.29C22, (C2×C4).23Q16
►Regular action on 128 points
Generators in S128
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 87)(26 88)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(49 100)(50 101)(51 102)(52 103)(53 104)(54 97)(55 98)(56 99)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 121)(105 118)(106 119)(107 120)(108 113)(109 114)(110 115)(111 116)(112 117)
(1 125 87 49)(2 101 88 93)(3 127 81 51)(4 103 82 95)(5 121 83 53)(6 97 84 89)(7 123 85 55)(8 99 86 91)(9 67 45 115)(10 111 46 79)(11 69 47 117)(12 105 48 73)(13 71 41 119)(14 107 42 75)(15 65 43 113)(16 109 44 77)(17 72 33 120)(18 108 34 76)(19 66 35 114)(20 110 36 78)(21 68 37 116)(22 112 38 80)(23 70 39 118)(24 106 40 74)(25 100 57 92)(26 126 58 50)(27 102 59 94)(28 128 60 52)(29 104 61 96)(30 122 62 54)(31 98 63 90)(32 124 64 56)
(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 43 83 11)(2 17 84 37)(3 41 85 9)(4 23 86 35)(5 47 87 15)(6 21 88 33)(7 45 81 13)(8 19 82 39)(10 58 42 30)(12 64 44 28)(14 62 46 26)(16 60 48 32)(18 29 38 57)(20 27 40 63)(22 25 34 61)(24 31 36 59)(49 76 121 112)(50 72 122 116)(51 74 123 110)(52 70 124 114)(53 80 125 108)(54 68 126 120)(55 78 127 106)(56 66 128 118)(65 96 117 100)(67 94 119 98)(69 92 113 104)(71 90 115 102)(73 91 109 103)(75 89 111 101)(77 95 105 99)(79 93 107 97)
G:=sub<Sym(128)| (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(49,100)(50,101)(51,102)(52,103)(53,104)(54,97)(55,98)(56,99)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,121)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,125,87,49)(2,101,88,93)(3,127,81,51)(4,103,82,95)(5,121,83,53)(6,97,84,89)(7,123,85,55)(8,99,86,91)(9,67,45,115)(10,111,46,79)(11,69,47,117)(12,105,48,73)(13,71,41,119)(14,107,42,75)(15,65,43,113)(16,109,44,77)(17,72,33,120)(18,108,34,76)(19,66,35,114)(20,110,36,78)(21,68,37,116)(22,112,38,80)(23,70,39,118)(24,106,40,74)(25,100,57,92)(26,126,58,50)(27,102,59,94)(28,128,60,52)(29,104,61,96)(30,122,62,54)(31,98,63,90)(32,124,64,56), (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), (1,43,83,11)(2,17,84,37)(3,41,85,9)(4,23,86,35)(5,47,87,15)(6,21,88,33)(7,45,81,13)(8,19,82,39)(10,58,42,30)(12,64,44,28)(14,62,46,26)(16,60,48,32)(18,29,38,57)(20,27,40,63)(22,25,34,61)(24,31,36,59)(49,76,121,112)(50,72,122,116)(51,74,123,110)(52,70,124,114)(53,80,125,108)(54,68,126,120)(55,78,127,106)(56,66,128,118)(65,96,117,100)(67,94,119,98)(69,92,113,104)(71,90,115,102)(73,91,109,103)(75,89,111,101)(77,95,105,99)(79,93,107,97)>;
G:=Group( (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(49,100)(50,101)(51,102)(52,103)(53,104)(54,97)(55,98)(56,99)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,121)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,125,87,49)(2,101,88,93)(3,127,81,51)(4,103,82,95)(5,121,83,53)(6,97,84,89)(7,123,85,55)(8,99,86,91)(9,67,45,115)(10,111,46,79)(11,69,47,117)(12,105,48,73)(13,71,41,119)(14,107,42,75)(15,65,43,113)(16,109,44,77)(17,72,33,120)(18,108,34,76)(19,66,35,114)(20,110,36,78)(21,68,37,116)(22,112,38,80)(23,70,39,118)(24,106,40,74)(25,100,57,92)(26,126,58,50)(27,102,59,94)(28,128,60,52)(29,104,61,96)(30,122,62,54)(31,98,63,90)(32,124,64,56), (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), (1,43,83,11)(2,17,84,37)(3,41,85,9)(4,23,86,35)(5,47,87,15)(6,21,88,33)(7,45,81,13)(8,19,82,39)(10,58,42,30)(12,64,44,28)(14,62,46,26)(16,60,48,32)(18,29,38,57)(20,27,40,63)(22,25,34,61)(24,31,36,59)(49,76,121,112)(50,72,122,116)(51,74,123,110)(52,70,124,114)(53,80,125,108)(54,68,126,120)(55,78,127,106)(56,66,128,118)(65,96,117,100)(67,94,119,98)(69,92,113,104)(71,90,115,102)(73,91,109,103)(75,89,111,101)(77,95,105,99)(79,93,107,97) );
G=PermutationGroup([[(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,87),(26,88),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(49,100),(50,101),(51,102),(52,103),(53,104),(54,97),(55,98),(56,99),(65,76),(66,77),(67,78),(68,79),(69,80),(70,73),(71,74),(72,75),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,121),(105,118),(106,119),(107,120),(108,113),(109,114),(110,115),(111,116),(112,117)], [(1,125,87,49),(2,101,88,93),(3,127,81,51),(4,103,82,95),(5,121,83,53),(6,97,84,89),(7,123,85,55),(8,99,86,91),(9,67,45,115),(10,111,46,79),(11,69,47,117),(12,105,48,73),(13,71,41,119),(14,107,42,75),(15,65,43,113),(16,109,44,77),(17,72,33,120),(18,108,34,76),(19,66,35,114),(20,110,36,78),(21,68,37,116),(22,112,38,80),(23,70,39,118),(24,106,40,74),(25,100,57,92),(26,126,58,50),(27,102,59,94),(28,128,60,52),(29,104,61,96),(30,122,62,54),(31,98,63,90),(32,124,64,56)], [(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)], [(1,43,83,11),(2,17,84,37),(3,41,85,9),(4,23,86,35),(5,47,87,15),(6,21,88,33),(7,45,81,13),(8,19,82,39),(10,58,42,30),(12,64,44,28),(14,62,46,26),(16,60,48,32),(18,29,38,57),(20,27,40,63),(22,25,34,61),(24,31,36,59),(49,76,121,112),(50,72,122,116),(51,74,123,110),(52,70,124,114),(53,80,125,108),(54,68,126,120),(55,78,127,106),(56,66,128,118),(65,96,117,100),(67,94,119,98),(69,92,113,104),(71,90,115,102),(73,91,109,103),(75,89,111,101),(77,95,105,99),(79,93,107,97)]])
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 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | SD16 | Q16 | C4○D4 | C8⋊C22 |
| kernel | (C2×C4).23Q16 | C22.7C42 | C22.4Q16 | C42⋊9C4 | C23.65C23 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 2 |
Matrix representation of (C2×C4).23Q16 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 5 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,8,1,0,0,0,0,3,9,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[14,3,0,0,0,0,3,3,0,0,0,0,0,0,13,13,0,0,0,0,8,4,0,0,0,0,0,0,0,5,0,0,0,0,7,7],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,0,0,0,0,1] >;
(C2×C4).23Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{23}Q_{16} % in TeX
G:=Group("(C2xC4).23Q16"); // GroupNames label
G:=SmallGroup(128,832);
// by ID
G=gap.SmallGroup(128,832);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,624,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2*c^4,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*c*d^-1=a*c^-1>;
// generators/relations