p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.123D4, C4⋊C4.18Q8, C4.31(C4×Q8), C4.32(C4⋊Q8), C42.C2⋊8C4, C2.7(D4.Q8), C2.7(Q8.Q8), C42.165(C2×C4), C23.812(C2×D4), (C22×C4).701D4, C4.51(C4.4D4), C22.79(C4○D8), (C22×C8).63C22, C22.4Q16.18C2, (C2×C42).335C22, C22.84(C22⋊Q8), C22.101(C8⋊C22), (C22×C4).1423C23, C22.90(C8.C22), C2.30(C23.36D4), C2.30(C23.24D4), C2.11(C23.67C23), (C4×C4⋊C4).29C2, (C2×C4⋊C8).35C2, C4⋊C4.96(C2×C4), (C2×C4).278(C2×Q8), (C2×C4).1363(C2×D4), (C2×C42.C2).5C2, (C2×C4).768(C4○D4), (C2×C4⋊C4).782C22, (C2×C4).437(C22×C4), (C2×C4).206(C22⋊C4), C22.298(C2×C22⋊C4), SmallGroup(128,721)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.123D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2b-1c-1 >
Subgroups: 236 in 130 conjugacy classes, 64 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C22×C8, C22.4Q16, C4×C4⋊C4, C2×C4⋊C8, C2×C42.C2, C42.123D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C4○D8, C8⋊C22, C8.C22, C23.67C23, C23.24D4, C23.36D4, D4.Q8, Q8.Q8, C42.123D4
►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 29 12 34)(2 30 9 35)(3 31 10 36)(4 32 11 33)(5 123 23 106)(6 124 24 107)(7 121 21 108)(8 122 22 105)(13 39 25 52)(14 40 26 49)(15 37 27 50)(16 38 28 51)(17 118 126 109)(18 119 127 110)(19 120 128 111)(20 117 125 112)(41 62 54 71)(42 63 55 72)(43 64 56 69)(44 61 53 70)(45 84 57 67)(46 81 58 68)(47 82 59 65)(48 83 60 66)(73 100 90 113)(74 97 91 114)(75 98 92 115)(76 99 89 116)(77 94 86 103)(78 95 87 104)(79 96 88 101)(80 93 85 102)
(1 82 50 53)(2 83 51 54)(3 84 52 55)(4 81 49 56)(5 95 112 76)(6 96 109 73)(7 93 110 74)(8 94 111 75)(9 66 38 41)(10 67 39 42)(11 68 40 43)(12 65 37 44)(13 63 31 45)(14 64 32 46)(15 61 29 47)(16 62 30 48)(17 113 124 79)(18 114 121 80)(19 115 122 77)(20 116 123 78)(21 102 119 91)(22 103 120 92)(23 104 117 89)(24 101 118 90)(25 72 36 57)(26 69 33 58)(27 70 34 59)(28 71 35 60)(85 127 97 108)(86 128 98 105)(87 125 99 106)(88 126 100 107)
(1 108 3 106)(2 124 4 122)(5 34 7 36)(6 32 8 30)(9 107 11 105)(10 123 12 121)(13 117 15 119)(14 111 16 109)(17 49 19 51)(18 39 20 37)(21 31 23 29)(22 35 24 33)(25 112 27 110)(26 120 28 118)(38 126 40 128)(41 103 43 101)(42 93 44 95)(45 114 47 116)(46 100 48 98)(50 127 52 125)(53 104 55 102)(54 94 56 96)(57 97 59 99)(58 113 60 115)(61 78 63 80)(62 86 64 88)(65 76 67 74)(66 92 68 90)(69 79 71 77)(70 87 72 85)(73 83 75 81)(82 89 84 91)
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), (1,29,12,34)(2,30,9,35)(3,31,10,36)(4,32,11,33)(5,123,23,106)(6,124,24,107)(7,121,21,108)(8,122,22,105)(13,39,25,52)(14,40,26,49)(15,37,27,50)(16,38,28,51)(17,118,126,109)(18,119,127,110)(19,120,128,111)(20,117,125,112)(41,62,54,71)(42,63,55,72)(43,64,56,69)(44,61,53,70)(45,84,57,67)(46,81,58,68)(47,82,59,65)(48,83,60,66)(73,100,90,113)(74,97,91,114)(75,98,92,115)(76,99,89,116)(77,94,86,103)(78,95,87,104)(79,96,88,101)(80,93,85,102), (1,82,50,53)(2,83,51,54)(3,84,52,55)(4,81,49,56)(5,95,112,76)(6,96,109,73)(7,93,110,74)(8,94,111,75)(9,66,38,41)(10,67,39,42)(11,68,40,43)(12,65,37,44)(13,63,31,45)(14,64,32,46)(15,61,29,47)(16,62,30,48)(17,113,124,79)(18,114,121,80)(19,115,122,77)(20,116,123,78)(21,102,119,91)(22,103,120,92)(23,104,117,89)(24,101,118,90)(25,72,36,57)(26,69,33,58)(27,70,34,59)(28,71,35,60)(85,127,97,108)(86,128,98,105)(87,125,99,106)(88,126,100,107), (1,108,3,106)(2,124,4,122)(5,34,7,36)(6,32,8,30)(9,107,11,105)(10,123,12,121)(13,117,15,119)(14,111,16,109)(17,49,19,51)(18,39,20,37)(21,31,23,29)(22,35,24,33)(25,112,27,110)(26,120,28,118)(38,126,40,128)(41,103,43,101)(42,93,44,95)(45,114,47,116)(46,100,48,98)(50,127,52,125)(53,104,55,102)(54,94,56,96)(57,97,59,99)(58,113,60,115)(61,78,63,80)(62,86,64,88)(65,76,67,74)(66,92,68,90)(69,79,71,77)(70,87,72,85)(73,83,75,81)(82,89,84,91)>;
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), (1,29,12,34)(2,30,9,35)(3,31,10,36)(4,32,11,33)(5,123,23,106)(6,124,24,107)(7,121,21,108)(8,122,22,105)(13,39,25,52)(14,40,26,49)(15,37,27,50)(16,38,28,51)(17,118,126,109)(18,119,127,110)(19,120,128,111)(20,117,125,112)(41,62,54,71)(42,63,55,72)(43,64,56,69)(44,61,53,70)(45,84,57,67)(46,81,58,68)(47,82,59,65)(48,83,60,66)(73,100,90,113)(74,97,91,114)(75,98,92,115)(76,99,89,116)(77,94,86,103)(78,95,87,104)(79,96,88,101)(80,93,85,102), (1,82,50,53)(2,83,51,54)(3,84,52,55)(4,81,49,56)(5,95,112,76)(6,96,109,73)(7,93,110,74)(8,94,111,75)(9,66,38,41)(10,67,39,42)(11,68,40,43)(12,65,37,44)(13,63,31,45)(14,64,32,46)(15,61,29,47)(16,62,30,48)(17,113,124,79)(18,114,121,80)(19,115,122,77)(20,116,123,78)(21,102,119,91)(22,103,120,92)(23,104,117,89)(24,101,118,90)(25,72,36,57)(26,69,33,58)(27,70,34,59)(28,71,35,60)(85,127,97,108)(86,128,98,105)(87,125,99,106)(88,126,100,107), (1,108,3,106)(2,124,4,122)(5,34,7,36)(6,32,8,30)(9,107,11,105)(10,123,12,121)(13,117,15,119)(14,111,16,109)(17,49,19,51)(18,39,20,37)(21,31,23,29)(22,35,24,33)(25,112,27,110)(26,120,28,118)(38,126,40,128)(41,103,43,101)(42,93,44,95)(45,114,47,116)(46,100,48,98)(50,127,52,125)(53,104,55,102)(54,94,56,96)(57,97,59,99)(58,113,60,115)(61,78,63,80)(62,86,64,88)(65,76,67,74)(66,92,68,90)(69,79,71,77)(70,87,72,85)(73,83,75,81)(82,89,84,91) );
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)], [(1,29,12,34),(2,30,9,35),(3,31,10,36),(4,32,11,33),(5,123,23,106),(6,124,24,107),(7,121,21,108),(8,122,22,105),(13,39,25,52),(14,40,26,49),(15,37,27,50),(16,38,28,51),(17,118,126,109),(18,119,127,110),(19,120,128,111),(20,117,125,112),(41,62,54,71),(42,63,55,72),(43,64,56,69),(44,61,53,70),(45,84,57,67),(46,81,58,68),(47,82,59,65),(48,83,60,66),(73,100,90,113),(74,97,91,114),(75,98,92,115),(76,99,89,116),(77,94,86,103),(78,95,87,104),(79,96,88,101),(80,93,85,102)], [(1,82,50,53),(2,83,51,54),(3,84,52,55),(4,81,49,56),(5,95,112,76),(6,96,109,73),(7,93,110,74),(8,94,111,75),(9,66,38,41),(10,67,39,42),(11,68,40,43),(12,65,37,44),(13,63,31,45),(14,64,32,46),(15,61,29,47),(16,62,30,48),(17,113,124,79),(18,114,121,80),(19,115,122,77),(20,116,123,78),(21,102,119,91),(22,103,120,92),(23,104,117,89),(24,101,118,90),(25,72,36,57),(26,69,33,58),(27,70,34,59),(28,71,35,60),(85,127,97,108),(86,128,98,105),(87,125,99,106),(88,126,100,107)], [(1,108,3,106),(2,124,4,122),(5,34,7,36),(6,32,8,30),(9,107,11,105),(10,123,12,121),(13,117,15,119),(14,111,16,109),(17,49,19,51),(18,39,20,37),(21,31,23,29),(22,35,24,33),(25,112,27,110),(26,120,28,118),(38,126,40,128),(41,103,43,101),(42,93,44,95),(45,114,47,116),(46,100,48,98),(50,127,52,125),(53,104,55,102),(54,94,56,96),(57,97,59,99),(58,113,60,115),(61,78,63,80),(62,86,64,88),(65,76,67,74),(66,92,68,90),(69,79,71,77),(70,87,72,85),(73,83,75,81),(82,89,84,91)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C42.123D4 | C22.4Q16 | C4×C4⋊C4 | C2×C4⋊C8 | C2×C42.C2 | C42.C2 | C42 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 8 | 1 | 1 |
Matrix representation of C42.123D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 2 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 6 | 14 | 0 | 0 | 0 | 0 |
| 6 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 0 | 1 | 7 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,2,0,0,0,0,16,16,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[6,6,0,0,0,0,14,11,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,10,1,0,0,0,0,1,7] >;
C42.123D4 in GAP, Magma, Sage, TeX
C_4^2._{123}D_4 % in TeX
G:=Group("C4^2.123D4"); // GroupNames label
G:=SmallGroup(128,721);
// by ID
G=gap.SmallGroup(128,721);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,436,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^-1*c^-1>;
// generators/relations