p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.111D4, C4⋊Q8⋊17C4, C4.13(C4×D4), (C2×C8).205D4, C4.76(C4⋊D4), C42.155(C2×C4), C2.3(C8.2D4), (C22×C4).297D4, C23.799(C2×D4), C42⋊8C4.11C2, C22.35(C4⋊1D4), (C22×C8).404C22, (C2×C42).316C22, (C22×Q8).35C22, (C22×C4).1404C23, C22.66(C4.4D4), C22.84(C8.C22), C2.22(C23.38D4), C2.12(C24.3C22), C2.3(C42.30C22), (C2×C4⋊Q8).10C2, (C2×C4).738(C2×D4), (C2×Q8).91(C2×C4), (C2×C8⋊C4).33C2, (C2×C4⋊C4).87C22, (C2×C4).595(C4○D4), (C2×C4).418(C22×C4), (C2×Q8⋊C4).34C2, (C2×C4).137(C22⋊C4), C22.282(C2×C22⋊C4), SmallGroup(128,692)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.111D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 292 in 154 conjugacy classes, 64 normal (14 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, C2.C42, C8⋊C4, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C42⋊8C4, C2×C8⋊C4, C2×Q8⋊C4, C2×C4⋊Q8, C42.111D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C8.C22, C24.3C22, C23.38D4, C42.30C22, C8.2D4, C42.111D4
►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 52 12 38)(2 49 9 39)(3 50 10 40)(4 51 11 37)(5 111 23 124)(6 112 24 121)(7 109 21 122)(8 110 22 123)(13 29 25 35)(14 30 26 36)(15 31 27 33)(16 32 28 34)(17 105 126 117)(18 106 127 118)(19 107 128 119)(20 108 125 120)(41 83 45 66)(42 84 46 67)(43 81 47 68)(44 82 48 65)(53 61 60 70)(54 62 57 71)(55 63 58 72)(56 64 59 69)(73 104 90 94)(74 101 91 95)(75 102 92 96)(76 103 89 93)(77 99 86 113)(78 100 87 114)(79 97 88 115)(80 98 85 116)
(1 95 35 87)(2 104 36 77)(3 93 33 85)(4 102 34 79)(5 67 118 60)(6 83 119 56)(7 65 120 58)(8 81 117 54)(9 94 30 86)(10 103 31 80)(11 96 32 88)(12 101 29 78)(13 100 52 91)(14 113 49 73)(15 98 50 89)(16 115 51 75)(17 71 110 43)(18 61 111 46)(19 69 112 41)(20 63 109 48)(21 82 108 55)(22 68 105 57)(23 84 106 53)(24 66 107 59)(25 114 38 74)(26 99 39 90)(27 116 40 76)(28 97 37 92)(42 127 70 124)(44 125 72 122)(45 128 64 121)(47 126 62 123)
(1 124 12 111)(2 123 9 110)(3 122 10 109)(4 121 11 112)(5 52 23 38)(6 51 24 37)(7 50 21 40)(8 49 22 39)(13 106 25 118)(14 105 26 117)(15 108 27 120)(16 107 28 119)(17 36 126 30)(18 35 127 29)(19 34 128 32)(20 33 125 31)(41 115 45 97)(42 114 46 100)(43 113 47 99)(44 116 48 98)(53 95 60 101)(54 94 57 104)(55 93 58 103)(56 96 59 102)(61 91 70 74)(62 90 71 73)(63 89 72 76)(64 92 69 75)(65 80 82 85)(66 79 83 88)(67 78 84 87)(68 77 81 86)
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), (1,52,12,38)(2,49,9,39)(3,50,10,40)(4,51,11,37)(5,111,23,124)(6,112,24,121)(7,109,21,122)(8,110,22,123)(13,29,25,35)(14,30,26,36)(15,31,27,33)(16,32,28,34)(17,105,126,117)(18,106,127,118)(19,107,128,119)(20,108,125,120)(41,83,45,66)(42,84,46,67)(43,81,47,68)(44,82,48,65)(53,61,60,70)(54,62,57,71)(55,63,58,72)(56,64,59,69)(73,104,90,94)(74,101,91,95)(75,102,92,96)(76,103,89,93)(77,99,86,113)(78,100,87,114)(79,97,88,115)(80,98,85,116), (1,95,35,87)(2,104,36,77)(3,93,33,85)(4,102,34,79)(5,67,118,60)(6,83,119,56)(7,65,120,58)(8,81,117,54)(9,94,30,86)(10,103,31,80)(11,96,32,88)(12,101,29,78)(13,100,52,91)(14,113,49,73)(15,98,50,89)(16,115,51,75)(17,71,110,43)(18,61,111,46)(19,69,112,41)(20,63,109,48)(21,82,108,55)(22,68,105,57)(23,84,106,53)(24,66,107,59)(25,114,38,74)(26,99,39,90)(27,116,40,76)(28,97,37,92)(42,127,70,124)(44,125,72,122)(45,128,64,121)(47,126,62,123), (1,124,12,111)(2,123,9,110)(3,122,10,109)(4,121,11,112)(5,52,23,38)(6,51,24,37)(7,50,21,40)(8,49,22,39)(13,106,25,118)(14,105,26,117)(15,108,27,120)(16,107,28,119)(17,36,126,30)(18,35,127,29)(19,34,128,32)(20,33,125,31)(41,115,45,97)(42,114,46,100)(43,113,47,99)(44,116,48,98)(53,95,60,101)(54,94,57,104)(55,93,58,103)(56,96,59,102)(61,91,70,74)(62,90,71,73)(63,89,72,76)(64,92,69,75)(65,80,82,85)(66,79,83,88)(67,78,84,87)(68,77,81,86)>;
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), (1,52,12,38)(2,49,9,39)(3,50,10,40)(4,51,11,37)(5,111,23,124)(6,112,24,121)(7,109,21,122)(8,110,22,123)(13,29,25,35)(14,30,26,36)(15,31,27,33)(16,32,28,34)(17,105,126,117)(18,106,127,118)(19,107,128,119)(20,108,125,120)(41,83,45,66)(42,84,46,67)(43,81,47,68)(44,82,48,65)(53,61,60,70)(54,62,57,71)(55,63,58,72)(56,64,59,69)(73,104,90,94)(74,101,91,95)(75,102,92,96)(76,103,89,93)(77,99,86,113)(78,100,87,114)(79,97,88,115)(80,98,85,116), (1,95,35,87)(2,104,36,77)(3,93,33,85)(4,102,34,79)(5,67,118,60)(6,83,119,56)(7,65,120,58)(8,81,117,54)(9,94,30,86)(10,103,31,80)(11,96,32,88)(12,101,29,78)(13,100,52,91)(14,113,49,73)(15,98,50,89)(16,115,51,75)(17,71,110,43)(18,61,111,46)(19,69,112,41)(20,63,109,48)(21,82,108,55)(22,68,105,57)(23,84,106,53)(24,66,107,59)(25,114,38,74)(26,99,39,90)(27,116,40,76)(28,97,37,92)(42,127,70,124)(44,125,72,122)(45,128,64,121)(47,126,62,123), (1,124,12,111)(2,123,9,110)(3,122,10,109)(4,121,11,112)(5,52,23,38)(6,51,24,37)(7,50,21,40)(8,49,22,39)(13,106,25,118)(14,105,26,117)(15,108,27,120)(16,107,28,119)(17,36,126,30)(18,35,127,29)(19,34,128,32)(20,33,125,31)(41,115,45,97)(42,114,46,100)(43,113,47,99)(44,116,48,98)(53,95,60,101)(54,94,57,104)(55,93,58,103)(56,96,59,102)(61,91,70,74)(62,90,71,73)(63,89,72,76)(64,92,69,75)(65,80,82,85)(66,79,83,88)(67,78,84,87)(68,77,81,86) );
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)], [(1,52,12,38),(2,49,9,39),(3,50,10,40),(4,51,11,37),(5,111,23,124),(6,112,24,121),(7,109,21,122),(8,110,22,123),(13,29,25,35),(14,30,26,36),(15,31,27,33),(16,32,28,34),(17,105,126,117),(18,106,127,118),(19,107,128,119),(20,108,125,120),(41,83,45,66),(42,84,46,67),(43,81,47,68),(44,82,48,65),(53,61,60,70),(54,62,57,71),(55,63,58,72),(56,64,59,69),(73,104,90,94),(74,101,91,95),(75,102,92,96),(76,103,89,93),(77,99,86,113),(78,100,87,114),(79,97,88,115),(80,98,85,116)], [(1,95,35,87),(2,104,36,77),(3,93,33,85),(4,102,34,79),(5,67,118,60),(6,83,119,56),(7,65,120,58),(8,81,117,54),(9,94,30,86),(10,103,31,80),(11,96,32,88),(12,101,29,78),(13,100,52,91),(14,113,49,73),(15,98,50,89),(16,115,51,75),(17,71,110,43),(18,61,111,46),(19,69,112,41),(20,63,109,48),(21,82,108,55),(22,68,105,57),(23,84,106,53),(24,66,107,59),(25,114,38,74),(26,99,39,90),(27,116,40,76),(28,97,37,92),(42,127,70,124),(44,125,72,122),(45,128,64,121),(47,126,62,123)], [(1,124,12,111),(2,123,9,110),(3,122,10,109),(4,121,11,112),(5,52,23,38),(6,51,24,37),(7,50,21,40),(8,49,22,39),(13,106,25,118),(14,105,26,117),(15,108,27,120),(16,107,28,119),(17,36,126,30),(18,35,127,29),(19,34,128,32),(20,33,125,31),(41,115,45,97),(42,114,46,100),(43,113,47,99),(44,116,48,98),(53,95,60,101),(54,94,57,104),(55,93,58,103),(56,96,59,102),(61,91,70,74),(62,90,71,73),(63,89,72,76),(64,92,69,75),(65,80,82,85),(66,79,83,88),(67,78,84,87),(68,77,81,86)]])
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 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C8.C22 |
| kernel | C42.111D4 | C42⋊8C4 | C2×C8⋊C4 | C2×Q8⋊C4 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C2×C8 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 8 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C42.111D4 ►in GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 14 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 10 | 12 | 10 |
| 0 | 0 | 0 | 0 | 10 | 14 | 10 | 5 |
| 0 | 0 | 0 | 0 | 5 | 7 | 3 | 10 |
| 0 | 0 | 0 | 0 | 7 | 12 | 10 | 14 |
| 3 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 15 | 11 |
| 0 | 0 | 0 | 0 | 3 | 16 | 11 | 2 |
| 0 | 0 | 0 | 0 | 15 | 11 | 16 | 14 |
| 0 | 0 | 0 | 0 | 11 | 2 | 14 | 1 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[14,10,0,0,0,0,0,0,16,3,0,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,5,15,0,0,0,0,0,0,0,0,3,10,5,7,0,0,0,0,10,14,7,12,0,0,0,0,12,10,3,10,0,0,0,0,10,5,10,14],[3,7,0,0,0,0,0,0,11,14,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,8,10,0,0,0,0,0,0,0,0,1,3,15,11,0,0,0,0,3,16,11,2,0,0,0,0,15,11,16,14,0,0,0,0,11,2,14,1] >;
C42.111D4 in GAP, Magma, Sage, TeX
C_4^2._{111}D_4 % in TeX
G:=Group("C4^2.111D4"); // GroupNames label
G:=SmallGroup(128,692);
// by ID
G=gap.SmallGroup(128,692);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,436,2019,248]);
// 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=a^-1*b^2,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