p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.8Q8, C42.29D4, C4⋊C8⋊7C4, (C2×C4).116D8, (C2×C4).8C42, (C2×C4).50Q16, C4.1(C2.D8), C4.1(C4.Q8), C42.36(C2×C4), (C2×C4).90SD16, C42⋊9C4.1C2, C2.2(C4.D8), (C22×C4).722D4, C4.30(D4⋊C4), C4.22(Q8⋊C4), C2.2(C4.10D8), C2.2(C4.6Q16), C2.7(C22.4Q16), (C2×C42).133C22, C22.39(D4⋊C4), C2.7(C22.C42), C23.216(C22⋊C4), C22.29(Q8⋊C4), C22.25(C4.D4), C22.16(C4.10D4), C22.46(C2.C42), (C2×C4⋊C4).5C4, (C2×C4⋊C8).6C2, (C2×C4).96(C4⋊C4), (C22×C4).151(C2×C4), (C2×C4).339(C22⋊C4), SmallGroup(128,28)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.8Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a-1b2c3 >
Subgroups: 184 in 94 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C4⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C42⋊9C4, C2×C4⋊C8, C42.8Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, C4.D4, C4.10D4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C4.D8, C4.10D8, C4.6Q16, C22.4Q16, C22.C42, C42.8Q8
►Regular action on 128 points
Generators in S128
(1 21 43 61)(2 62 44 22)(3 23 45 63)(4 64 46 24)(5 17 47 57)(6 58 48 18)(7 19 41 59)(8 60 42 20)(9 55 87 27)(10 28 88 56)(11 49 81 29)(12 30 82 50)(13 51 83 31)(14 32 84 52)(15 53 85 25)(16 26 86 54)(33 66 100 79)(34 80 101 67)(35 68 102 73)(36 74 103 69)(37 70 104 75)(38 76 97 71)(39 72 98 77)(40 78 99 65)(89 113 123 108)(90 109 124 114)(91 115 125 110)(92 111 126 116)(93 117 127 112)(94 105 128 118)(95 119 121 106)(96 107 122 120)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(33 123 37 127)(34 124 38 128)(35 125 39 121)(36 126 40 122)(41 51 45 55)(42 52 46 56)(43 53 47 49)(44 54 48 50)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)(65 107 69 111)(66 108 70 112)(67 109 71 105)(68 110 72 106)(73 115 77 119)(74 116 78 120)(75 117 79 113)(76 118 80 114)(89 104 93 100)(90 97 94 101)(91 98 95 102)(92 99 96 103)
(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 35 31 123)(2 80 32 120)(3 33 25 121)(4 78 26 118)(5 39 27 127)(6 76 28 116)(7 37 29 125)(8 74 30 114)(9 117 17 77)(10 92 18 97)(11 115 19 75)(12 90 20 103)(13 113 21 73)(14 96 22 101)(15 119 23 79)(16 94 24 99)(34 84 122 62)(36 82 124 60)(38 88 126 58)(40 86 128 64)(41 104 49 91)(42 69 50 109)(43 102 51 89)(44 67 52 107)(45 100 53 95)(46 65 54 105)(47 98 55 93)(48 71 56 111)(57 72 87 112)(59 70 81 110)(61 68 83 108)(63 66 85 106)
G:=sub<Sym(128)| (1,21,43,61)(2,62,44,22)(3,23,45,63)(4,64,46,24)(5,17,47,57)(6,58,48,18)(7,19,41,59)(8,60,42,20)(9,55,87,27)(10,28,88,56)(11,49,81,29)(12,30,82,50)(13,51,83,31)(14,32,84,52)(15,53,85,25)(16,26,86,54)(33,66,100,79)(34,80,101,67)(35,68,102,73)(36,74,103,69)(37,70,104,75)(38,76,97,71)(39,72,98,77)(40,78,99,65)(89,113,123,108)(90,109,124,114)(91,115,125,110)(92,111,126,116)(93,117,127,112)(94,105,128,118)(95,119,121,106)(96,107,122,120), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(33,123,37,127)(34,124,38,128)(35,125,39,121)(36,126,40,122)(41,51,45,55)(42,52,46,56)(43,53,47,49)(44,54,48,50)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88)(65,107,69,111)(66,108,70,112)(67,109,71,105)(68,110,72,106)(73,115,77,119)(74,116,78,120)(75,117,79,113)(76,118,80,114)(89,104,93,100)(90,97,94,101)(91,98,95,102)(92,99,96,103), (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), (1,35,31,123)(2,80,32,120)(3,33,25,121)(4,78,26,118)(5,39,27,127)(6,76,28,116)(7,37,29,125)(8,74,30,114)(9,117,17,77)(10,92,18,97)(11,115,19,75)(12,90,20,103)(13,113,21,73)(14,96,22,101)(15,119,23,79)(16,94,24,99)(34,84,122,62)(36,82,124,60)(38,88,126,58)(40,86,128,64)(41,104,49,91)(42,69,50,109)(43,102,51,89)(44,67,52,107)(45,100,53,95)(46,65,54,105)(47,98,55,93)(48,71,56,111)(57,72,87,112)(59,70,81,110)(61,68,83,108)(63,66,85,106)>;
G:=Group( (1,21,43,61)(2,62,44,22)(3,23,45,63)(4,64,46,24)(5,17,47,57)(6,58,48,18)(7,19,41,59)(8,60,42,20)(9,55,87,27)(10,28,88,56)(11,49,81,29)(12,30,82,50)(13,51,83,31)(14,32,84,52)(15,53,85,25)(16,26,86,54)(33,66,100,79)(34,80,101,67)(35,68,102,73)(36,74,103,69)(37,70,104,75)(38,76,97,71)(39,72,98,77)(40,78,99,65)(89,113,123,108)(90,109,124,114)(91,115,125,110)(92,111,126,116)(93,117,127,112)(94,105,128,118)(95,119,121,106)(96,107,122,120), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(33,123,37,127)(34,124,38,128)(35,125,39,121)(36,126,40,122)(41,51,45,55)(42,52,46,56)(43,53,47,49)(44,54,48,50)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88)(65,107,69,111)(66,108,70,112)(67,109,71,105)(68,110,72,106)(73,115,77,119)(74,116,78,120)(75,117,79,113)(76,118,80,114)(89,104,93,100)(90,97,94,101)(91,98,95,102)(92,99,96,103), (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), (1,35,31,123)(2,80,32,120)(3,33,25,121)(4,78,26,118)(5,39,27,127)(6,76,28,116)(7,37,29,125)(8,74,30,114)(9,117,17,77)(10,92,18,97)(11,115,19,75)(12,90,20,103)(13,113,21,73)(14,96,22,101)(15,119,23,79)(16,94,24,99)(34,84,122,62)(36,82,124,60)(38,88,126,58)(40,86,128,64)(41,104,49,91)(42,69,50,109)(43,102,51,89)(44,67,52,107)(45,100,53,95)(46,65,54,105)(47,98,55,93)(48,71,56,111)(57,72,87,112)(59,70,81,110)(61,68,83,108)(63,66,85,106) );
G=PermutationGroup([[(1,21,43,61),(2,62,44,22),(3,23,45,63),(4,64,46,24),(5,17,47,57),(6,58,48,18),(7,19,41,59),(8,60,42,20),(9,55,87,27),(10,28,88,56),(11,49,81,29),(12,30,82,50),(13,51,83,31),(14,32,84,52),(15,53,85,25),(16,26,86,54),(33,66,100,79),(34,80,101,67),(35,68,102,73),(36,74,103,69),(37,70,104,75),(38,76,97,71),(39,72,98,77),(40,78,99,65),(89,113,123,108),(90,109,124,114),(91,115,125,110),(92,111,126,116),(93,117,127,112),(94,105,128,118),(95,119,121,106),(96,107,122,120)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18),(33,123,37,127),(34,124,38,128),(35,125,39,121),(36,126,40,122),(41,51,45,55),(42,52,46,56),(43,53,47,49),(44,54,48,50),(57,81,61,85),(58,82,62,86),(59,83,63,87),(60,84,64,88),(65,107,69,111),(66,108,70,112),(67,109,71,105),(68,110,72,106),(73,115,77,119),(74,116,78,120),(75,117,79,113),(76,118,80,114),(89,104,93,100),(90,97,94,101),(91,98,95,102),(92,99,96,103)], [(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)], [(1,35,31,123),(2,80,32,120),(3,33,25,121),(4,78,26,118),(5,39,27,127),(6,76,28,116),(7,37,29,125),(8,74,30,114),(9,117,17,77),(10,92,18,97),(11,115,19,75),(12,90,20,103),(13,113,21,73),(14,96,22,101),(15,119,23,79),(16,94,24,99),(34,84,122,62),(36,82,124,60),(38,88,126,58),(40,86,128,64),(41,104,49,91),(42,69,50,109),(43,102,51,89),(44,67,52,107),(45,100,53,95),(46,65,54,105),(47,98,55,93),(48,71,56,111),(57,72,87,112),(59,70,81,110),(61,68,83,108),(63,66,85,106)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8P |
| 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | C4.D4 | C4.10D4 |
| kernel | C42.8Q8 | C42⋊9C4 | C2×C4⋊C8 | C4⋊C8 | C2×C4⋊C4 | C42 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 8 | 4 | 1 | 1 | 2 | 4 | 8 | 4 | 1 | 1 |
Matrix representation of C42.8Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 10 | 9 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 7 | 0 | 0 |
| 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 15 | 5 |
| 0 | 0 | 0 | 13 | 2 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,8,10,0,0,0,2,9],[4,0,0,0,0,0,1,7,0,0,0,7,16,0,0,0,0,0,15,13,0,0,0,5,2] >;
C42.8Q8 in GAP, Magma, Sage, TeX
C_4^2._8Q_8
% in TeX
G:=Group("C4^2.8Q8"); // GroupNames label
G:=SmallGroup(128,28);
// by ID
G=gap.SmallGroup(128,28);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,3924,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^-1*b^2*c^3>;
// generators/relations