p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.85D4, (C2×Q8).6Q8, C2.8(Q8.Q8), (C22×C4).144D4, C23.908(C2×D4), C4.32(C22⋊Q8), C4.33(C4.4D4), (C22×C8).68C22, C2.7(C23⋊Q8), C2.30(D4.7D4), C22.210C22≀C2, C22.106(C4○D8), C22.4Q16.22C2, (C2×C42).356C22, (C22×Q8).61C22, (C22×C4).1442C23, C22.89(C4.4D4), C22.105(C22⋊Q8), C22.123(C8.C22), C22.7C42.11C2, C23.67C23.14C2, C2.5(C42.30C22), C2.7(C42.78C22), (C2×C4).282(C2×Q8), (C2×C4).1032(C2×D4), (C2×C42.C2).9C2, (C2×C4).619(C4○D4), (C2×C4⋊C4).115C22, (C2×Q8⋊C4).11C2, SmallGroup(128,758)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4.85D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 256 in 129 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C23.67C23, C2×Q8⋊C4, C2×C42.C2, C4⋊C4.85D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4.4D4, C4○D8, C8.C22, C23⋊Q8, D4.7D4, Q8.Q8, C42.78C22, C42.30C22, C4⋊C4.85D4
►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 66 8 76)(2 65 5 75)(3 68 6 74)(4 67 7 73)(9 85 22 79)(10 88 23 78)(11 87 24 77)(12 86 21 80)(13 69 126 62)(14 72 127 61)(15 71 128 64)(16 70 125 63)(17 91 27 82)(18 90 28 81)(19 89 25 84)(20 92 26 83)(29 104 39 95)(30 103 40 94)(31 102 37 93)(32 101 38 96)(33 108 42 98)(34 107 43 97)(35 106 44 100)(36 105 41 99)(45 117 54 111)(46 120 55 110)(47 119 56 109)(48 118 53 112)(49 123 59 114)(50 122 60 113)(51 121 57 116)(52 124 58 115)
(1 62 10 60)(2 61 11 59)(3 64 12 58)(4 63 9 57)(5 72 24 49)(6 71 21 52)(7 70 22 51)(8 69 23 50)(13 87 113 65)(14 86 114 68)(15 85 115 67)(16 88 116 66)(17 54 39 36)(18 53 40 35)(19 56 37 34)(20 55 38 33)(25 47 31 43)(26 46 32 42)(27 45 29 41)(28 48 30 44)(73 128 79 124)(74 127 80 123)(75 126 77 122)(76 125 78 121)(81 117 103 99)(82 120 104 98)(83 119 101 97)(84 118 102 100)(89 112 93 106)(90 111 94 105)(91 110 95 108)(92 109 96 107)
(1 35 6 42)(2 36 7 43)(3 33 8 44)(4 34 5 41)(9 56 24 45)(10 53 21 46)(11 54 22 47)(12 55 23 48)(13 94 128 101)(14 95 125 102)(15 96 126 103)(16 93 127 104)(17 57 25 49)(18 58 26 50)(19 59 27 51)(20 60 28 52)(29 70 37 61)(30 71 38 62)(31 72 39 63)(32 69 40 64)(65 99 73 107)(66 100 74 108)(67 97 75 105)(68 98 76 106)(77 111 85 119)(78 112 86 120)(79 109 87 117)(80 110 88 118)(81 115 92 122)(82 116 89 123)(83 113 90 124)(84 114 91 121)
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), (1,66,8,76)(2,65,5,75)(3,68,6,74)(4,67,7,73)(9,85,22,79)(10,88,23,78)(11,87,24,77)(12,86,21,80)(13,69,126,62)(14,72,127,61)(15,71,128,64)(16,70,125,63)(17,91,27,82)(18,90,28,81)(19,89,25,84)(20,92,26,83)(29,104,39,95)(30,103,40,94)(31,102,37,93)(32,101,38,96)(33,108,42,98)(34,107,43,97)(35,106,44,100)(36,105,41,99)(45,117,54,111)(46,120,55,110)(47,119,56,109)(48,118,53,112)(49,123,59,114)(50,122,60,113)(51,121,57,116)(52,124,58,115), (1,62,10,60)(2,61,11,59)(3,64,12,58)(4,63,9,57)(5,72,24,49)(6,71,21,52)(7,70,22,51)(8,69,23,50)(13,87,113,65)(14,86,114,68)(15,85,115,67)(16,88,116,66)(17,54,39,36)(18,53,40,35)(19,56,37,34)(20,55,38,33)(25,47,31,43)(26,46,32,42)(27,45,29,41)(28,48,30,44)(73,128,79,124)(74,127,80,123)(75,126,77,122)(76,125,78,121)(81,117,103,99)(82,120,104,98)(83,119,101,97)(84,118,102,100)(89,112,93,106)(90,111,94,105)(91,110,95,108)(92,109,96,107), (1,35,6,42)(2,36,7,43)(3,33,8,44)(4,34,5,41)(9,56,24,45)(10,53,21,46)(11,54,22,47)(12,55,23,48)(13,94,128,101)(14,95,125,102)(15,96,126,103)(16,93,127,104)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(29,70,37,61)(30,71,38,62)(31,72,39,63)(32,69,40,64)(65,99,73,107)(66,100,74,108)(67,97,75,105)(68,98,76,106)(77,111,85,119)(78,112,86,120)(79,109,87,117)(80,110,88,118)(81,115,92,122)(82,116,89,123)(83,113,90,124)(84,114,91,121)>;
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), (1,66,8,76)(2,65,5,75)(3,68,6,74)(4,67,7,73)(9,85,22,79)(10,88,23,78)(11,87,24,77)(12,86,21,80)(13,69,126,62)(14,72,127,61)(15,71,128,64)(16,70,125,63)(17,91,27,82)(18,90,28,81)(19,89,25,84)(20,92,26,83)(29,104,39,95)(30,103,40,94)(31,102,37,93)(32,101,38,96)(33,108,42,98)(34,107,43,97)(35,106,44,100)(36,105,41,99)(45,117,54,111)(46,120,55,110)(47,119,56,109)(48,118,53,112)(49,123,59,114)(50,122,60,113)(51,121,57,116)(52,124,58,115), (1,62,10,60)(2,61,11,59)(3,64,12,58)(4,63,9,57)(5,72,24,49)(6,71,21,52)(7,70,22,51)(8,69,23,50)(13,87,113,65)(14,86,114,68)(15,85,115,67)(16,88,116,66)(17,54,39,36)(18,53,40,35)(19,56,37,34)(20,55,38,33)(25,47,31,43)(26,46,32,42)(27,45,29,41)(28,48,30,44)(73,128,79,124)(74,127,80,123)(75,126,77,122)(76,125,78,121)(81,117,103,99)(82,120,104,98)(83,119,101,97)(84,118,102,100)(89,112,93,106)(90,111,94,105)(91,110,95,108)(92,109,96,107), (1,35,6,42)(2,36,7,43)(3,33,8,44)(4,34,5,41)(9,56,24,45)(10,53,21,46)(11,54,22,47)(12,55,23,48)(13,94,128,101)(14,95,125,102)(15,96,126,103)(16,93,127,104)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(29,70,37,61)(30,71,38,62)(31,72,39,63)(32,69,40,64)(65,99,73,107)(66,100,74,108)(67,97,75,105)(68,98,76,106)(77,111,85,119)(78,112,86,120)(79,109,87,117)(80,110,88,118)(81,115,92,122)(82,116,89,123)(83,113,90,124)(84,114,91,121) );
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)], [(1,66,8,76),(2,65,5,75),(3,68,6,74),(4,67,7,73),(9,85,22,79),(10,88,23,78),(11,87,24,77),(12,86,21,80),(13,69,126,62),(14,72,127,61),(15,71,128,64),(16,70,125,63),(17,91,27,82),(18,90,28,81),(19,89,25,84),(20,92,26,83),(29,104,39,95),(30,103,40,94),(31,102,37,93),(32,101,38,96),(33,108,42,98),(34,107,43,97),(35,106,44,100),(36,105,41,99),(45,117,54,111),(46,120,55,110),(47,119,56,109),(48,118,53,112),(49,123,59,114),(50,122,60,113),(51,121,57,116),(52,124,58,115)], [(1,62,10,60),(2,61,11,59),(3,64,12,58),(4,63,9,57),(5,72,24,49),(6,71,21,52),(7,70,22,51),(8,69,23,50),(13,87,113,65),(14,86,114,68),(15,85,115,67),(16,88,116,66),(17,54,39,36),(18,53,40,35),(19,56,37,34),(20,55,38,33),(25,47,31,43),(26,46,32,42),(27,45,29,41),(28,48,30,44),(73,128,79,124),(74,127,80,123),(75,126,77,122),(76,125,78,121),(81,117,103,99),(82,120,104,98),(83,119,101,97),(84,118,102,100),(89,112,93,106),(90,111,94,105),(91,110,95,108),(92,109,96,107)], [(1,35,6,42),(2,36,7,43),(3,33,8,44),(4,34,5,41),(9,56,24,45),(10,53,21,46),(11,54,22,47),(12,55,23,48),(13,94,128,101),(14,95,125,102),(15,96,126,103),(16,93,127,104),(17,57,25,49),(18,58,26,50),(19,59,27,51),(20,60,28,52),(29,70,37,61),(30,71,38,62),(31,72,39,63),(32,69,40,64),(65,99,73,107),(66,100,74,108),(67,97,75,105),(68,98,76,106),(77,111,85,119),(78,112,86,120),(79,109,87,117),(80,110,88,118),(81,115,92,122),(82,116,89,123),(83,113,90,124),(84,114,91,121)]])
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 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C4○D8 | C8.C22 |
| kernel | C4⋊C4.85D4 | C22.7C42 | C22.4Q16 | C23.67C23 | C2×Q8⋊C4 | C2×C42.C2 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 6 | 8 | 2 |
Matrix representation of C4⋊C4.85D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 15 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 12 | 15 | 0 | 0 | 0 | 0 |
| 13 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 12 | 0 | 0 |
| 0 | 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 14 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[12,12,0,0,0,0,15,5,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[12,13,0,0,0,0,15,5,0,0,0,0,0,0,14,5,0,0,0,0,12,3,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[5,5,0,0,0,0,2,12,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,0,13,0,0,0,0,4,0] >;
C4⋊C4.85D4 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{85}D_4 % in TeX
G:=Group("C4:C4.85D4"); // GroupNames label
G:=SmallGroup(128,758);
// by ID
G=gap.SmallGroup(128,758);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,512,422,387,352,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations