p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.117D4, C4⋊Q8⋊21C4, C4.95(C4×D4), C4⋊C4.225D4, (C2×C4).62Q16, C4⋊1(Q8⋊C4), C4.23(C4⋊1D4), C42.159(C2×C4), C2.5(C4⋊2Q16), (C2×C4).105SD16, (C22×C4).762D4, C23.807(C2×D4), C22.41(C2×Q16), C2.5(D4.D4), C4.66(C4.4D4), (C22×C8).56C22, C22.73(C2×SD16), (C2×C42).327C22, (C22×Q8).44C22, C22.152(C4⋊D4), (C22×C4).1417C23, C22.87(C8.C22), C2.23(C23.38D4), C2.33(C24.3C22), (C2×C4⋊C8).30C2, (C4×C4⋊C4).25C2, (C2×C4⋊Q8).12C2, (C2×C4).1359(C2×D4), (C2×Q8⋊C4).9C2, (C2×Q8).104(C2×C4), C2.23(C2×Q8⋊C4), (C2×C4).869(C4○D4), (C2×C4⋊C4).777C22, (C2×C4).431(C22×C4), (C2×C4).259(C22⋊C4), C22.291(C2×C22⋊C4), SmallGroup(128,713)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.117D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 300 in 162 conjugacy classes, 72 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C4×C4⋊C4, C2×Q8⋊C4, C2×C4⋊C8, C2×C4⋊Q8, C42.117D4
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, C8.C22, C24.3C22, C2×Q8⋊C4, C23.38D4, D4.D4, C4⋊2Q16, C42.117D4
►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 51 11 25)(2 52 12 26)(3 49 9 27)(4 50 10 28)(5 126 118 122)(6 127 119 123)(7 128 120 124)(8 125 117 121)(13 33 39 31)(14 34 40 32)(15 35 37 29)(16 36 38 30)(17 112 108 23)(18 109 105 24)(19 110 106 21)(20 111 107 22)(41 67 69 45)(42 68 70 46)(43 65 71 47)(44 66 72 48)(53 83 61 57)(54 84 62 58)(55 81 63 59)(56 82 64 60)(73 77 101 99)(74 78 102 100)(75 79 103 97)(76 80 104 98)(85 93 115 89)(86 94 116 90)(87 95 113 91)(88 96 114 92)
(1 45 13 61)(2 46 14 62)(3 47 15 63)(4 48 16 64)(5 78 19 94)(6 79 20 95)(7 80 17 96)(8 77 18 93)(9 65 37 55)(10 66 38 56)(11 67 39 53)(12 68 40 54)(21 116 122 102)(22 113 123 103)(23 114 124 104)(24 115 121 101)(25 41 31 57)(26 42 32 58)(27 43 29 59)(28 44 30 60)(33 83 51 69)(34 84 52 70)(35 81 49 71)(36 82 50 72)(73 109 85 125)(74 110 86 126)(75 111 87 127)(76 112 88 128)(89 117 99 105)(90 118 100 106)(91 119 97 107)(92 120 98 108)
(1 107 11 20)(2 106 12 19)(3 105 9 18)(4 108 10 17)(5 14 118 40)(6 13 119 39)(7 16 120 38)(8 15 117 37)(21 52 110 26)(22 51 111 25)(23 50 112 28)(24 49 109 27)(29 121 35 125)(30 124 36 128)(31 123 33 127)(32 122 34 126)(41 79 69 97)(42 78 70 100)(43 77 71 99)(44 80 72 98)(45 103 67 75)(46 102 68 74)(47 101 65 73)(48 104 66 76)(53 87 61 113)(54 86 62 116)(55 85 63 115)(56 88 64 114)(57 95 83 91)(58 94 84 90)(59 93 81 89)(60 96 82 92)
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), (1,51,11,25)(2,52,12,26)(3,49,9,27)(4,50,10,28)(5,126,118,122)(6,127,119,123)(7,128,120,124)(8,125,117,121)(13,33,39,31)(14,34,40,32)(15,35,37,29)(16,36,38,30)(17,112,108,23)(18,109,105,24)(19,110,106,21)(20,111,107,22)(41,67,69,45)(42,68,70,46)(43,65,71,47)(44,66,72,48)(53,83,61,57)(54,84,62,58)(55,81,63,59)(56,82,64,60)(73,77,101,99)(74,78,102,100)(75,79,103,97)(76,80,104,98)(85,93,115,89)(86,94,116,90)(87,95,113,91)(88,96,114,92), (1,45,13,61)(2,46,14,62)(3,47,15,63)(4,48,16,64)(5,78,19,94)(6,79,20,95)(7,80,17,96)(8,77,18,93)(9,65,37,55)(10,66,38,56)(11,67,39,53)(12,68,40,54)(21,116,122,102)(22,113,123,103)(23,114,124,104)(24,115,121,101)(25,41,31,57)(26,42,32,58)(27,43,29,59)(28,44,30,60)(33,83,51,69)(34,84,52,70)(35,81,49,71)(36,82,50,72)(73,109,85,125)(74,110,86,126)(75,111,87,127)(76,112,88,128)(89,117,99,105)(90,118,100,106)(91,119,97,107)(92,120,98,108), (1,107,11,20)(2,106,12,19)(3,105,9,18)(4,108,10,17)(5,14,118,40)(6,13,119,39)(7,16,120,38)(8,15,117,37)(21,52,110,26)(22,51,111,25)(23,50,112,28)(24,49,109,27)(29,121,35,125)(30,124,36,128)(31,123,33,127)(32,122,34,126)(41,79,69,97)(42,78,70,100)(43,77,71,99)(44,80,72,98)(45,103,67,75)(46,102,68,74)(47,101,65,73)(48,104,66,76)(53,87,61,113)(54,86,62,116)(55,85,63,115)(56,88,64,114)(57,95,83,91)(58,94,84,90)(59,93,81,89)(60,96,82,92)>;
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), (1,51,11,25)(2,52,12,26)(3,49,9,27)(4,50,10,28)(5,126,118,122)(6,127,119,123)(7,128,120,124)(8,125,117,121)(13,33,39,31)(14,34,40,32)(15,35,37,29)(16,36,38,30)(17,112,108,23)(18,109,105,24)(19,110,106,21)(20,111,107,22)(41,67,69,45)(42,68,70,46)(43,65,71,47)(44,66,72,48)(53,83,61,57)(54,84,62,58)(55,81,63,59)(56,82,64,60)(73,77,101,99)(74,78,102,100)(75,79,103,97)(76,80,104,98)(85,93,115,89)(86,94,116,90)(87,95,113,91)(88,96,114,92), (1,45,13,61)(2,46,14,62)(3,47,15,63)(4,48,16,64)(5,78,19,94)(6,79,20,95)(7,80,17,96)(8,77,18,93)(9,65,37,55)(10,66,38,56)(11,67,39,53)(12,68,40,54)(21,116,122,102)(22,113,123,103)(23,114,124,104)(24,115,121,101)(25,41,31,57)(26,42,32,58)(27,43,29,59)(28,44,30,60)(33,83,51,69)(34,84,52,70)(35,81,49,71)(36,82,50,72)(73,109,85,125)(74,110,86,126)(75,111,87,127)(76,112,88,128)(89,117,99,105)(90,118,100,106)(91,119,97,107)(92,120,98,108), (1,107,11,20)(2,106,12,19)(3,105,9,18)(4,108,10,17)(5,14,118,40)(6,13,119,39)(7,16,120,38)(8,15,117,37)(21,52,110,26)(22,51,111,25)(23,50,112,28)(24,49,109,27)(29,121,35,125)(30,124,36,128)(31,123,33,127)(32,122,34,126)(41,79,69,97)(42,78,70,100)(43,77,71,99)(44,80,72,98)(45,103,67,75)(46,102,68,74)(47,101,65,73)(48,104,66,76)(53,87,61,113)(54,86,62,116)(55,85,63,115)(56,88,64,114)(57,95,83,91)(58,94,84,90)(59,93,81,89)(60,96,82,92) );
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)], [(1,51,11,25),(2,52,12,26),(3,49,9,27),(4,50,10,28),(5,126,118,122),(6,127,119,123),(7,128,120,124),(8,125,117,121),(13,33,39,31),(14,34,40,32),(15,35,37,29),(16,36,38,30),(17,112,108,23),(18,109,105,24),(19,110,106,21),(20,111,107,22),(41,67,69,45),(42,68,70,46),(43,65,71,47),(44,66,72,48),(53,83,61,57),(54,84,62,58),(55,81,63,59),(56,82,64,60),(73,77,101,99),(74,78,102,100),(75,79,103,97),(76,80,104,98),(85,93,115,89),(86,94,116,90),(87,95,113,91),(88,96,114,92)], [(1,45,13,61),(2,46,14,62),(3,47,15,63),(4,48,16,64),(5,78,19,94),(6,79,20,95),(7,80,17,96),(8,77,18,93),(9,65,37,55),(10,66,38,56),(11,67,39,53),(12,68,40,54),(21,116,122,102),(22,113,123,103),(23,114,124,104),(24,115,121,101),(25,41,31,57),(26,42,32,58),(27,43,29,59),(28,44,30,60),(33,83,51,69),(34,84,52,70),(35,81,49,71),(36,82,50,72),(73,109,85,125),(74,110,86,126),(75,111,87,127),(76,112,88,128),(89,117,99,105),(90,118,100,106),(91,119,97,107),(92,120,98,108)], [(1,107,11,20),(2,106,12,19),(3,105,9,18),(4,108,10,17),(5,14,118,40),(6,13,119,39),(7,16,120,38),(8,15,117,37),(21,52,110,26),(22,51,111,25),(23,50,112,28),(24,49,109,27),(29,121,35,125),(30,124,36,128),(31,123,33,127),(32,122,34,126),(41,79,69,97),(42,78,70,100),(43,77,71,99),(44,80,72,98),(45,103,67,75),(46,102,68,74),(47,101,65,73),(48,104,66,76),(53,87,61,113),(54,86,62,116),(55,85,63,115),(56,88,64,114),(57,95,83,91),(58,94,84,90),(59,93,81,89),(60,96,82,92)]])
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 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | SD16 | Q16 | C4○D4 | C8.C22 |
| kernel | C42.117D4 | C4×C4⋊C4 | C2×Q8⋊C4 | C2×C4⋊C8 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C42.117D4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 15 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 9 | 0 | 0 |
| 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 8 | 14 |
| 0 | 0 | 0 | 16 | 9 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 6 | 14 |
| 0 | 0 | 0 | 1 | 11 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,15,1],[13,0,0,0,0,0,4,4,0,0,0,9,13,0,0,0,0,0,8,16,0,0,0,14,9],[16,0,0,0,0,0,1,1,0,0,0,0,16,0,0,0,0,0,6,1,0,0,0,14,11] >;
C42.117D4 in GAP, Magma, Sage, TeX
C_4^2._{117}D_4 % in TeX
G:=Group("C4^2.117D4"); // GroupNames label
G:=SmallGroup(128,713);
// by ID
G=gap.SmallGroup(128,713);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,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