p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.436D4, C4⋊Q8⋊25C4, (C2×C8)⋊11Q8, (C2×C4).76D8, C4.12(C4×Q8), (C2×C4).38Q16, C2.3(C8⋊2Q8), C2.3(C8⋊3Q8), (C2×C4).74SD16, C22.50(C2×D8), C42.269(C2×C4), C2.4(C4.4D8), C23.813(C2×D4), (C22×C4).581D4, C4.80(C22⋊Q8), C22.35(C4⋊Q8), C4.22(D4⋊C4), C22.43(C2×Q16), C42⋊9C4.12C2, C4.19(Q8⋊C4), C2.4(C4.SD16), C22.77(C2×SD16), C22.4Q16.19C2, (C22×C8).496C22, (C2×C42).1079C22, (C22×C4).1424C23, C22.68(C4.4D4), C2.12(C23.67C23), (C2×C4×C8).25C2, C4⋊C4.97(C2×C4), (C2×C4⋊Q8).15C2, (C2×C4).211(C2×Q8), C2.25(C2×D4⋊C4), (C2×C4).1364(C2×D4), (C2×C4⋊C4).95C22, C2.25(C2×Q8⋊C4), (C2×C4).607(C4○D4), (C2×C4).438(C22×C4), (C2×C4).264(C22⋊C4), C22.299(C2×C22⋊C4), SmallGroup(128,722)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.436D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 292 in 150 conjugacy classes, 80 normal (26 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C4×C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C22.4Q16, C42⋊9C4, C2×C4×C8, C2×C4⋊Q8, C42.436D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C23.67C23, C2×D4⋊C4, C2×Q8⋊C4, C4.4D8, C4.SD16, C8⋊3Q8, C8⋊2Q8, C42.436D4
►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 11 13)(2 40 12 14)(3 37 9 15)(4 38 10 16)(5 106 118 19)(6 107 119 20)(7 108 120 17)(8 105 117 18)(21 126 110 122)(22 127 111 123)(23 128 112 124)(24 125 109 121)(25 31 51 33)(26 32 52 34)(27 29 49 35)(28 30 50 36)(41 83 69 57)(42 84 70 58)(43 81 71 59)(44 82 72 60)(45 61 67 53)(46 62 68 54)(47 63 65 55)(48 64 66 56)(73 85 101 115)(74 86 102 116)(75 87 103 113)(76 88 104 114)(77 89 99 93)(78 90 100 94)(79 91 97 95)(80 92 98 96)
(1 87 25 95)(2 86 26 94)(3 85 27 93)(4 88 28 96)(5 84 122 54)(6 83 123 53)(7 82 124 56)(8 81 121 55)(9 115 49 89)(10 114 50 92)(11 113 51 91)(12 116 52 90)(13 103 33 79)(14 102 34 78)(15 101 35 77)(16 104 36 80)(17 72 112 48)(18 71 109 47)(19 70 110 46)(20 69 111 45)(21 68 106 42)(22 67 107 41)(23 66 108 44)(24 65 105 43)(29 99 37 73)(30 98 38 76)(31 97 39 75)(32 100 40 74)(57 127 61 119)(58 126 62 118)(59 125 63 117)(60 128 64 120)
(1 121 9 127)(2 124 10 126)(3 123 11 125)(4 122 12 128)(5 52 120 28)(6 51 117 27)(7 50 118 26)(8 49 119 25)(13 24 37 111)(14 23 38 110)(15 22 39 109)(16 21 40 112)(17 36 106 32)(18 35 107 31)(19 34 108 30)(20 33 105 29)(41 113 71 85)(42 116 72 88)(43 115 69 87)(44 114 70 86)(45 95 65 89)(46 94 66 92)(47 93 67 91)(48 96 68 90)(53 79 63 99)(54 78 64 98)(55 77 61 97)(56 80 62 100)(57 75 81 101)(58 74 82 104)(59 73 83 103)(60 76 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), (1,39,11,13)(2,40,12,14)(3,37,9,15)(4,38,10,16)(5,106,118,19)(6,107,119,20)(7,108,120,17)(8,105,117,18)(21,126,110,122)(22,127,111,123)(23,128,112,124)(24,125,109,121)(25,31,51,33)(26,32,52,34)(27,29,49,35)(28,30,50,36)(41,83,69,57)(42,84,70,58)(43,81,71,59)(44,82,72,60)(45,61,67,53)(46,62,68,54)(47,63,65,55)(48,64,66,56)(73,85,101,115)(74,86,102,116)(75,87,103,113)(76,88,104,114)(77,89,99,93)(78,90,100,94)(79,91,97,95)(80,92,98,96), (1,87,25,95)(2,86,26,94)(3,85,27,93)(4,88,28,96)(5,84,122,54)(6,83,123,53)(7,82,124,56)(8,81,121,55)(9,115,49,89)(10,114,50,92)(11,113,51,91)(12,116,52,90)(13,103,33,79)(14,102,34,78)(15,101,35,77)(16,104,36,80)(17,72,112,48)(18,71,109,47)(19,70,110,46)(20,69,111,45)(21,68,106,42)(22,67,107,41)(23,66,108,44)(24,65,105,43)(29,99,37,73)(30,98,38,76)(31,97,39,75)(32,100,40,74)(57,127,61,119)(58,126,62,118)(59,125,63,117)(60,128,64,120), (1,121,9,127)(2,124,10,126)(3,123,11,125)(4,122,12,128)(5,52,120,28)(6,51,117,27)(7,50,118,26)(8,49,119,25)(13,24,37,111)(14,23,38,110)(15,22,39,109)(16,21,40,112)(17,36,106,32)(18,35,107,31)(19,34,108,30)(20,33,105,29)(41,113,71,85)(42,116,72,88)(43,115,69,87)(44,114,70,86)(45,95,65,89)(46,94,66,92)(47,93,67,91)(48,96,68,90)(53,79,63,99)(54,78,64,98)(55,77,61,97)(56,80,62,100)(57,75,81,101)(58,74,82,104)(59,73,83,103)(60,76,84,102)>;
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), (1,39,11,13)(2,40,12,14)(3,37,9,15)(4,38,10,16)(5,106,118,19)(6,107,119,20)(7,108,120,17)(8,105,117,18)(21,126,110,122)(22,127,111,123)(23,128,112,124)(24,125,109,121)(25,31,51,33)(26,32,52,34)(27,29,49,35)(28,30,50,36)(41,83,69,57)(42,84,70,58)(43,81,71,59)(44,82,72,60)(45,61,67,53)(46,62,68,54)(47,63,65,55)(48,64,66,56)(73,85,101,115)(74,86,102,116)(75,87,103,113)(76,88,104,114)(77,89,99,93)(78,90,100,94)(79,91,97,95)(80,92,98,96), (1,87,25,95)(2,86,26,94)(3,85,27,93)(4,88,28,96)(5,84,122,54)(6,83,123,53)(7,82,124,56)(8,81,121,55)(9,115,49,89)(10,114,50,92)(11,113,51,91)(12,116,52,90)(13,103,33,79)(14,102,34,78)(15,101,35,77)(16,104,36,80)(17,72,112,48)(18,71,109,47)(19,70,110,46)(20,69,111,45)(21,68,106,42)(22,67,107,41)(23,66,108,44)(24,65,105,43)(29,99,37,73)(30,98,38,76)(31,97,39,75)(32,100,40,74)(57,127,61,119)(58,126,62,118)(59,125,63,117)(60,128,64,120), (1,121,9,127)(2,124,10,126)(3,123,11,125)(4,122,12,128)(5,52,120,28)(6,51,117,27)(7,50,118,26)(8,49,119,25)(13,24,37,111)(14,23,38,110)(15,22,39,109)(16,21,40,112)(17,36,106,32)(18,35,107,31)(19,34,108,30)(20,33,105,29)(41,113,71,85)(42,116,72,88)(43,115,69,87)(44,114,70,86)(45,95,65,89)(46,94,66,92)(47,93,67,91)(48,96,68,90)(53,79,63,99)(54,78,64,98)(55,77,61,97)(56,80,62,100)(57,75,81,101)(58,74,82,104)(59,73,83,103)(60,76,84,102) );
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)], [(1,39,11,13),(2,40,12,14),(3,37,9,15),(4,38,10,16),(5,106,118,19),(6,107,119,20),(7,108,120,17),(8,105,117,18),(21,126,110,122),(22,127,111,123),(23,128,112,124),(24,125,109,121),(25,31,51,33),(26,32,52,34),(27,29,49,35),(28,30,50,36),(41,83,69,57),(42,84,70,58),(43,81,71,59),(44,82,72,60),(45,61,67,53),(46,62,68,54),(47,63,65,55),(48,64,66,56),(73,85,101,115),(74,86,102,116),(75,87,103,113),(76,88,104,114),(77,89,99,93),(78,90,100,94),(79,91,97,95),(80,92,98,96)], [(1,87,25,95),(2,86,26,94),(3,85,27,93),(4,88,28,96),(5,84,122,54),(6,83,123,53),(7,82,124,56),(8,81,121,55),(9,115,49,89),(10,114,50,92),(11,113,51,91),(12,116,52,90),(13,103,33,79),(14,102,34,78),(15,101,35,77),(16,104,36,80),(17,72,112,48),(18,71,109,47),(19,70,110,46),(20,69,111,45),(21,68,106,42),(22,67,107,41),(23,66,108,44),(24,65,105,43),(29,99,37,73),(30,98,38,76),(31,97,39,75),(32,100,40,74),(57,127,61,119),(58,126,62,118),(59,125,63,117),(60,128,64,120)], [(1,121,9,127),(2,124,10,126),(3,123,11,125),(4,122,12,128),(5,52,120,28),(6,51,117,27),(7,50,118,26),(8,49,119,25),(13,24,37,111),(14,23,38,110),(15,22,39,109),(16,21,40,112),(17,36,106,32),(18,35,107,31),(19,34,108,30),(20,33,105,29),(41,113,71,85),(42,116,72,88),(43,115,69,87),(44,114,70,86),(45,95,65,89),(46,94,66,92),(47,93,67,91),(48,96,68,90),(53,79,63,99),(54,78,64,98),(55,77,61,97),(56,80,62,100),(57,75,81,101),(58,74,82,104),(59,73,83,103),(60,76,84,102)]])
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 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | C4○D4 |
| kernel | C42.436D4 | C22.4Q16 | C42⋊9C4 | C2×C4×C8 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 8 | 4 | 4 |
Matrix representation of C42.436D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 16 | 1 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,16,0,0,0,2,1],[13,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,9,13],[1,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,0,5,0,0,0,10,0] >;
C42.436D4 in GAP, Magma, Sage, TeX
C_4^2._{436}D_4 % in TeX
G:=Group("C4^2.436D4"); // GroupNames label
G:=SmallGroup(128,722);
// by ID
G=gap.SmallGroup(128,722);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^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