p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.7Q8, C23.947(C2×D4), (C22×C4).166D4, C2.15(D4.Q8), C2.15(Q8.Q8), C42⋊8C4.18C2, (C22×C8).92C22, C4.27(C42.C2), C4.24(C42⋊2C2), C22.131(C4○D8), C22.4Q16.32C2, (C2×C42).398C22, C22.162(C8⋊C22), (C22×C4).1481C23, C22.99(C4.4D4), C22.124(C22⋊Q8), C2.15(C23.20D4), C2.15(C23.19D4), C22.151(C8.C22), C22.7C42.17C2, C23.65C23.28C2, C2.7(C23.83C23), C2.11(C42.28C22), C2.11(C42.78C22), C22.134(C22.D4), (C2×C4).298(C2×Q8), (C2×C4).632(C4○D4), (C2×C4⋊C4).170C22, SmallGroup(128,833)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4.Q8
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b-1, dbd-1=a-1b-1, dcd-1=a2c-1 >
Subgroups: 208 in 99 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C42⋊8C4, C23.65C23, C4⋊C4.Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4○D8, C8⋊C22, C8.C22, C23.83C23, D4.Q8, Q8.Q8, C23.19D4, C23.20D4, C42.78C22, C42.28C22, C4⋊C4.Q8
►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 81 9 90)(2 84 10 89)(3 83 11 92)(4 82 12 91)(5 101 21 94)(6 104 22 93)(7 103 23 96)(8 102 24 95)(13 53 127 45)(14 56 128 48)(15 55 125 47)(16 54 126 46)(17 65 26 73)(18 68 27 76)(19 67 28 75)(20 66 25 74)(29 78 40 85)(30 77 37 88)(31 80 38 87)(32 79 39 86)(33 122 42 114)(34 121 43 113)(35 124 44 116)(36 123 41 115)(49 111 57 119)(50 110 58 118)(51 109 59 117)(52 112 60 120)(61 98 69 107)(62 97 70 106)(63 100 71 105)(64 99 72 108)
(1 67 7 88)(2 68 8 85)(3 65 5 86)(4 66 6 87)(9 75 23 77)(10 76 24 78)(11 73 21 79)(12 74 22 80)(13 71 113 49)(14 72 114 50)(15 69 115 51)(16 70 116 52)(17 103 39 81)(18 104 40 82)(19 101 37 83)(20 102 38 84)(25 95 31 89)(26 96 32 90)(27 93 29 91)(28 94 30 92)(33 120 56 97)(34 117 53 98)(35 118 54 99)(36 119 55 100)(41 111 47 105)(42 112 48 106)(43 109 45 107)(44 110 46 108)(57 127 63 121)(58 128 64 122)(59 125 61 123)(60 126 62 124)
(1 41 23 55)(2 44 24 54)(3 43 21 53)(4 42 22 56)(5 45 11 34)(6 48 12 33)(7 47 9 36)(8 46 10 35)(13 84 121 95)(14 83 122 94)(15 82 123 93)(16 81 124 96)(17 64 32 50)(18 63 29 49)(19 62 30 52)(20 61 31 51)(25 69 38 59)(26 72 39 58)(27 71 40 57)(28 70 37 60)(65 105 79 119)(66 108 80 118)(67 107 77 117)(68 106 78 120)(73 100 86 111)(74 99 87 110)(75 98 88 109)(76 97 85 112)(89 113 102 127)(90 116 103 126)(91 115 104 125)(92 114 101 128)
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), (1,81,9,90)(2,84,10,89)(3,83,11,92)(4,82,12,91)(5,101,21,94)(6,104,22,93)(7,103,23,96)(8,102,24,95)(13,53,127,45)(14,56,128,48)(15,55,125,47)(16,54,126,46)(17,65,26,73)(18,68,27,76)(19,67,28,75)(20,66,25,74)(29,78,40,85)(30,77,37,88)(31,80,38,87)(32,79,39,86)(33,122,42,114)(34,121,43,113)(35,124,44,116)(36,123,41,115)(49,111,57,119)(50,110,58,118)(51,109,59,117)(52,112,60,120)(61,98,69,107)(62,97,70,106)(63,100,71,105)(64,99,72,108), (1,67,7,88)(2,68,8,85)(3,65,5,86)(4,66,6,87)(9,75,23,77)(10,76,24,78)(11,73,21,79)(12,74,22,80)(13,71,113,49)(14,72,114,50)(15,69,115,51)(16,70,116,52)(17,103,39,81)(18,104,40,82)(19,101,37,83)(20,102,38,84)(25,95,31,89)(26,96,32,90)(27,93,29,91)(28,94,30,92)(33,120,56,97)(34,117,53,98)(35,118,54,99)(36,119,55,100)(41,111,47,105)(42,112,48,106)(43,109,45,107)(44,110,46,108)(57,127,63,121)(58,128,64,122)(59,125,61,123)(60,126,62,124), (1,41,23,55)(2,44,24,54)(3,43,21,53)(4,42,22,56)(5,45,11,34)(6,48,12,33)(7,47,9,36)(8,46,10,35)(13,84,121,95)(14,83,122,94)(15,82,123,93)(16,81,124,96)(17,64,32,50)(18,63,29,49)(19,62,30,52)(20,61,31,51)(25,69,38,59)(26,72,39,58)(27,71,40,57)(28,70,37,60)(65,105,79,119)(66,108,80,118)(67,107,77,117)(68,106,78,120)(73,100,86,111)(74,99,87,110)(75,98,88,109)(76,97,85,112)(89,113,102,127)(90,116,103,126)(91,115,104,125)(92,114,101,128)>;
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), (1,81,9,90)(2,84,10,89)(3,83,11,92)(4,82,12,91)(5,101,21,94)(6,104,22,93)(7,103,23,96)(8,102,24,95)(13,53,127,45)(14,56,128,48)(15,55,125,47)(16,54,126,46)(17,65,26,73)(18,68,27,76)(19,67,28,75)(20,66,25,74)(29,78,40,85)(30,77,37,88)(31,80,38,87)(32,79,39,86)(33,122,42,114)(34,121,43,113)(35,124,44,116)(36,123,41,115)(49,111,57,119)(50,110,58,118)(51,109,59,117)(52,112,60,120)(61,98,69,107)(62,97,70,106)(63,100,71,105)(64,99,72,108), (1,67,7,88)(2,68,8,85)(3,65,5,86)(4,66,6,87)(9,75,23,77)(10,76,24,78)(11,73,21,79)(12,74,22,80)(13,71,113,49)(14,72,114,50)(15,69,115,51)(16,70,116,52)(17,103,39,81)(18,104,40,82)(19,101,37,83)(20,102,38,84)(25,95,31,89)(26,96,32,90)(27,93,29,91)(28,94,30,92)(33,120,56,97)(34,117,53,98)(35,118,54,99)(36,119,55,100)(41,111,47,105)(42,112,48,106)(43,109,45,107)(44,110,46,108)(57,127,63,121)(58,128,64,122)(59,125,61,123)(60,126,62,124), (1,41,23,55)(2,44,24,54)(3,43,21,53)(4,42,22,56)(5,45,11,34)(6,48,12,33)(7,47,9,36)(8,46,10,35)(13,84,121,95)(14,83,122,94)(15,82,123,93)(16,81,124,96)(17,64,32,50)(18,63,29,49)(19,62,30,52)(20,61,31,51)(25,69,38,59)(26,72,39,58)(27,71,40,57)(28,70,37,60)(65,105,79,119)(66,108,80,118)(67,107,77,117)(68,106,78,120)(73,100,86,111)(74,99,87,110)(75,98,88,109)(76,97,85,112)(89,113,102,127)(90,116,103,126)(91,115,104,125)(92,114,101,128) );
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)], [(1,81,9,90),(2,84,10,89),(3,83,11,92),(4,82,12,91),(5,101,21,94),(6,104,22,93),(7,103,23,96),(8,102,24,95),(13,53,127,45),(14,56,128,48),(15,55,125,47),(16,54,126,46),(17,65,26,73),(18,68,27,76),(19,67,28,75),(20,66,25,74),(29,78,40,85),(30,77,37,88),(31,80,38,87),(32,79,39,86),(33,122,42,114),(34,121,43,113),(35,124,44,116),(36,123,41,115),(49,111,57,119),(50,110,58,118),(51,109,59,117),(52,112,60,120),(61,98,69,107),(62,97,70,106),(63,100,71,105),(64,99,72,108)], [(1,67,7,88),(2,68,8,85),(3,65,5,86),(4,66,6,87),(9,75,23,77),(10,76,24,78),(11,73,21,79),(12,74,22,80),(13,71,113,49),(14,72,114,50),(15,69,115,51),(16,70,116,52),(17,103,39,81),(18,104,40,82),(19,101,37,83),(20,102,38,84),(25,95,31,89),(26,96,32,90),(27,93,29,91),(28,94,30,92),(33,120,56,97),(34,117,53,98),(35,118,54,99),(36,119,55,100),(41,111,47,105),(42,112,48,106),(43,109,45,107),(44,110,46,108),(57,127,63,121),(58,128,64,122),(59,125,61,123),(60,126,62,124)], [(1,41,23,55),(2,44,24,54),(3,43,21,53),(4,42,22,56),(5,45,11,34),(6,48,12,33),(7,47,9,36),(8,46,10,35),(13,84,121,95),(14,83,122,94),(15,82,123,93),(16,81,124,96),(17,64,32,50),(18,63,29,49),(19,62,30,52),(20,61,31,51),(25,69,38,59),(26,72,39,58),(27,71,40,57),(28,70,37,60),(65,105,79,119),(66,108,80,118),(67,107,77,117),(68,106,78,120),(73,100,86,111),(74,99,87,110),(75,98,88,109),(76,97,85,112),(89,113,102,127),(90,116,103,126),(91,115,104,125),(92,114,101,128)]])
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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C4⋊C4.Q8 | C22.7C42 | C22.4Q16 | C42⋊8C4 | C23.65C23 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 10 | 8 | 1 | 1 |
Matrix representation of C4⋊C4.Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 14 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 6 |
| 0 | 0 | 0 | 0 | 6 | 13 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 5 | 0 | 0 |
| 0 | 0 | 12 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 14 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 6 |
| 0 | 0 | 0 | 0 | 6 | 4 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,5,3,0,0,0,0,14,12,0,0,0,0,0,0,4,6,0,0,0,0,6,13],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,14,12,0,0,0,0,5,3,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,13,6,0,0,0,0,6,4] >;
C4⋊C4.Q8 in GAP, Magma, Sage, TeX
C_4\rtimes C_4.Q_8
% in TeX
G:=Group("C4:C4.Q8"); // GroupNames label
G:=SmallGroup(128,833);
// by ID
G=gap.SmallGroup(128,833);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,504,141,176,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2*c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations