p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.431D4, C4⋊Q8⋊16C4, C4.9(C4×D4), (C2×C8).229D4, (C2×C4).37Q16, C4⋊2(Q8⋊C4), C2.2(C8⋊5D4), (C2×C4).72SD16, C4.72(C4⋊D4), C42.265(C2×C4), C2.2(C4⋊Q16), C23.795(C2×D4), (C22×C4).579D4, C22.39(C2×Q16), C42⋊9C4.11C2, C2.3(C4.SD16), C22.70(C2×SD16), C22.31(C4⋊1D4), (C22×C8).489C22, (C22×Q8).32C22, (C22×C4).1400C23, (C2×C42).1071C22, C22.62(C4.4D4), C2.8(C24.3C22), (C2×C4×C8).23C2, (C2×C4⋊Q8).9C2, (C2×C4).734(C2×D4), (C2×Q8).88(C2×C4), (C2×C4⋊C4).83C22, (C2×Q8⋊C4).7C2, C2.22(C2×Q8⋊C4), (C2×C4).591(C4○D4), (C2×C4).414(C22×C4), (C2×C4).255(C22⋊C4), C22.278(C2×C22⋊C4), SmallGroup(128,688)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.431D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 308 in 166 conjugacy classes, 80 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C42⋊9C4, C2×C4×C8, C2×Q8⋊C4, C2×C4⋊Q8, C42.431D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4○D4, Q8⋊C4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×SD16, C2×Q16, C24.3C22, C2×Q8⋊C4, C4.SD16, C8⋊5D4, C4⋊Q16, C42.431D4
►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 39 29 13)(2 40 30 14)(3 37 31 15)(4 38 32 16)(5 120 108 19)(6 117 105 20)(7 118 106 17)(8 119 107 18)(9 49 35 25)(10 50 36 26)(11 51 33 27)(12 52 34 28)(21 110 122 126)(22 111 123 127)(23 112 124 128)(24 109 121 125)(41 81 71 57)(42 82 72 58)(43 83 69 59)(44 84 70 60)(45 53 67 61)(46 54 68 62)(47 55 65 63)(48 56 66 64)(73 103 113 85)(74 104 114 86)(75 101 115 87)(76 102 116 88)(77 89 93 99)(78 90 94 100)(79 91 95 97)(80 92 96 98)
(1 87 9 79)(2 86 10 78)(3 85 11 77)(4 88 12 80)(5 84 23 66)(6 83 24 65)(7 82 21 68)(8 81 22 67)(13 75 25 91)(14 74 26 90)(15 73 27 89)(16 76 28 92)(17 72 126 62)(18 71 127 61)(19 70 128 64)(20 69 125 63)(29 101 35 95)(30 104 36 94)(31 103 33 93)(32 102 34 96)(37 113 51 99)(38 116 52 98)(39 115 49 97)(40 114 50 100)(41 111 53 119)(42 110 54 118)(43 109 55 117)(44 112 56 120)(45 107 57 123)(46 106 58 122)(47 105 59 121)(48 108 60 124)
(1 127 29 111)(2 126 30 110)(3 125 31 109)(4 128 32 112)(5 52 108 28)(6 51 105 27)(7 50 106 26)(8 49 107 25)(9 18 35 119)(10 17 36 118)(11 20 33 117)(12 19 34 120)(13 22 39 123)(14 21 40 122)(15 24 37 121)(16 23 38 124)(41 115 71 75)(42 114 72 74)(43 113 69 73)(44 116 70 76)(45 79 67 95)(46 78 68 94)(47 77 65 93)(48 80 66 96)(53 97 61 91)(54 100 62 90)(55 99 63 89)(56 98 64 92)(57 87 81 101)(58 86 82 104)(59 85 83 103)(60 88 84 102)
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), (1,39,29,13)(2,40,30,14)(3,37,31,15)(4,38,32,16)(5,120,108,19)(6,117,105,20)(7,118,106,17)(8,119,107,18)(9,49,35,25)(10,50,36,26)(11,51,33,27)(12,52,34,28)(21,110,122,126)(22,111,123,127)(23,112,124,128)(24,109,121,125)(41,81,71,57)(42,82,72,58)(43,83,69,59)(44,84,70,60)(45,53,67,61)(46,54,68,62)(47,55,65,63)(48,56,66,64)(73,103,113,85)(74,104,114,86)(75,101,115,87)(76,102,116,88)(77,89,93,99)(78,90,94,100)(79,91,95,97)(80,92,96,98), (1,87,9,79)(2,86,10,78)(3,85,11,77)(4,88,12,80)(5,84,23,66)(6,83,24,65)(7,82,21,68)(8,81,22,67)(13,75,25,91)(14,74,26,90)(15,73,27,89)(16,76,28,92)(17,72,126,62)(18,71,127,61)(19,70,128,64)(20,69,125,63)(29,101,35,95)(30,104,36,94)(31,103,33,93)(32,102,34,96)(37,113,51,99)(38,116,52,98)(39,115,49,97)(40,114,50,100)(41,111,53,119)(42,110,54,118)(43,109,55,117)(44,112,56,120)(45,107,57,123)(46,106,58,122)(47,105,59,121)(48,108,60,124), (1,127,29,111)(2,126,30,110)(3,125,31,109)(4,128,32,112)(5,52,108,28)(6,51,105,27)(7,50,106,26)(8,49,107,25)(9,18,35,119)(10,17,36,118)(11,20,33,117)(12,19,34,120)(13,22,39,123)(14,21,40,122)(15,24,37,121)(16,23,38,124)(41,115,71,75)(42,114,72,74)(43,113,69,73)(44,116,70,76)(45,79,67,95)(46,78,68,94)(47,77,65,93)(48,80,66,96)(53,97,61,91)(54,100,62,90)(55,99,63,89)(56,98,64,92)(57,87,81,101)(58,86,82,104)(59,85,83,103)(60,88,84,102)>;
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), (1,39,29,13)(2,40,30,14)(3,37,31,15)(4,38,32,16)(5,120,108,19)(6,117,105,20)(7,118,106,17)(8,119,107,18)(9,49,35,25)(10,50,36,26)(11,51,33,27)(12,52,34,28)(21,110,122,126)(22,111,123,127)(23,112,124,128)(24,109,121,125)(41,81,71,57)(42,82,72,58)(43,83,69,59)(44,84,70,60)(45,53,67,61)(46,54,68,62)(47,55,65,63)(48,56,66,64)(73,103,113,85)(74,104,114,86)(75,101,115,87)(76,102,116,88)(77,89,93,99)(78,90,94,100)(79,91,95,97)(80,92,96,98), (1,87,9,79)(2,86,10,78)(3,85,11,77)(4,88,12,80)(5,84,23,66)(6,83,24,65)(7,82,21,68)(8,81,22,67)(13,75,25,91)(14,74,26,90)(15,73,27,89)(16,76,28,92)(17,72,126,62)(18,71,127,61)(19,70,128,64)(20,69,125,63)(29,101,35,95)(30,104,36,94)(31,103,33,93)(32,102,34,96)(37,113,51,99)(38,116,52,98)(39,115,49,97)(40,114,50,100)(41,111,53,119)(42,110,54,118)(43,109,55,117)(44,112,56,120)(45,107,57,123)(46,106,58,122)(47,105,59,121)(48,108,60,124), (1,127,29,111)(2,126,30,110)(3,125,31,109)(4,128,32,112)(5,52,108,28)(6,51,105,27)(7,50,106,26)(8,49,107,25)(9,18,35,119)(10,17,36,118)(11,20,33,117)(12,19,34,120)(13,22,39,123)(14,21,40,122)(15,24,37,121)(16,23,38,124)(41,115,71,75)(42,114,72,74)(43,113,69,73)(44,116,70,76)(45,79,67,95)(46,78,68,94)(47,77,65,93)(48,80,66,96)(53,97,61,91)(54,100,62,90)(55,99,63,89)(56,98,64,92)(57,87,81,101)(58,86,82,104)(59,85,83,103)(60,88,84,102) );
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)], [(1,39,29,13),(2,40,30,14),(3,37,31,15),(4,38,32,16),(5,120,108,19),(6,117,105,20),(7,118,106,17),(8,119,107,18),(9,49,35,25),(10,50,36,26),(11,51,33,27),(12,52,34,28),(21,110,122,126),(22,111,123,127),(23,112,124,128),(24,109,121,125),(41,81,71,57),(42,82,72,58),(43,83,69,59),(44,84,70,60),(45,53,67,61),(46,54,68,62),(47,55,65,63),(48,56,66,64),(73,103,113,85),(74,104,114,86),(75,101,115,87),(76,102,116,88),(77,89,93,99),(78,90,94,100),(79,91,95,97),(80,92,96,98)], [(1,87,9,79),(2,86,10,78),(3,85,11,77),(4,88,12,80),(5,84,23,66),(6,83,24,65),(7,82,21,68),(8,81,22,67),(13,75,25,91),(14,74,26,90),(15,73,27,89),(16,76,28,92),(17,72,126,62),(18,71,127,61),(19,70,128,64),(20,69,125,63),(29,101,35,95),(30,104,36,94),(31,103,33,93),(32,102,34,96),(37,113,51,99),(38,116,52,98),(39,115,49,97),(40,114,50,100),(41,111,53,119),(42,110,54,118),(43,109,55,117),(44,112,56,120),(45,107,57,123),(46,106,58,122),(47,105,59,121),(48,108,60,124)], [(1,127,29,111),(2,126,30,110),(3,125,31,109),(4,128,32,112),(5,52,108,28),(6,51,105,27),(7,50,106,26),(8,49,107,25),(9,18,35,119),(10,17,36,118),(11,20,33,117),(12,19,34,120),(13,22,39,123),(14,21,40,122),(15,24,37,121),(16,23,38,124),(41,115,71,75),(42,114,72,74),(43,113,69,73),(44,116,70,76),(45,79,67,95),(46,78,68,94),(47,77,65,93),(48,80,66,96),(53,97,61,91),(54,100,62,90),(55,99,63,89),(56,98,64,92),(57,87,81,101),(58,86,82,104),(59,85,83,103),(60,88,84,102)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | SD16 | Q16 | C4○D4 |
| kernel | C42.431D4 | C42⋊9C4 | C2×C4×C8 | C2×Q8⋊C4 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 8 | 2 | 4 | 2 | 8 | 8 | 4 |
Matrix representation of C42.431D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 6 | 0 | 0 |
| 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 8 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 11 | 0 | 0 |
| 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,13],[4,0,0,0,0,0,13,6,0,0,0,6,4,0,0,0,0,0,0,8,0,0,0,15,0],[16,0,0,0,0,0,4,11,0,0,0,11,13,0,0,0,0,0,0,4,0,0,0,4,0] >;
C42.431D4 in GAP, Magma, Sage, TeX
C_4^2._{431}D_4 % in TeX
G:=Group("C4^2.431D4"); // GroupNames label
G:=SmallGroup(128,688);
// by ID
G=gap.SmallGroup(128,688);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;
// generators/relations