p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊11Q8, C23.566C24, C22.2552- 1+4, C22.3402+ 1+4, C42⋊8C4.41C2, C42⋊5C4.13C2, (C2×C42).630C22, (C22×C4).171C23, C22.140(C22×Q8), (C22×Q8).171C22, C2.55(C22.32C24), C23.78C23.17C2, C2.C42.280C22, C23.63C23.39C2, C23.67C23.51C2, C23.65C23.70C2, C23.83C23.28C2, C23.81C23.30C2, C2.66(C22.36C24), C2.55(C22.33C24), C2.27(C23.41C23), C2.43(C23.37C23), C2.37(C22.35C24), (C4×C4⋊C4).79C2, (C2×C4).136(C2×Q8), (C2×C4).186(C4○D4), (C2×C4⋊C4).387C22, C22.433(C2×C4○D4), SmallGroup(128,1398)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊11Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c-1 >
Subgroups: 324 in 184 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C4×C4⋊C4, C42⋊8C4, C42⋊5C4, C23.63C23, C23.65C23, C23.67C23, C23.78C23, C23.81C23, C23.83C23, C42⋊11Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.37C23, C22.32C24, C22.33C24, C22.35C24, C22.36C24, C23.41C23, C42⋊11Q8
►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 13 42 79)(2 14 43 80)(3 15 44 77)(4 16 41 78)(5 100 72 34)(6 97 69 35)(7 98 70 36)(8 99 71 33)(9 105 75 46)(10 106 76 47)(11 107 73 48)(12 108 74 45)(17 115 83 56)(18 116 84 53)(19 113 81 54)(20 114 82 55)(21 52 87 111)(22 49 88 112)(23 50 85 109)(24 51 86 110)(25 121 91 62)(26 122 92 63)(27 123 89 64)(28 124 90 61)(29 58 95 117)(30 59 96 118)(31 60 93 119)(32 57 94 120)(37 67 103 127)(38 68 104 128)(39 65 101 125)(40 66 102 126)
(1 83 75 109)(2 18 76 51)(3 81 73 111)(4 20 74 49)(5 32 102 61)(6 95 103 121)(7 30 104 63)(8 93 101 123)(9 50 42 17)(10 110 43 84)(11 52 44 19)(12 112 41 82)(13 54 46 21)(14 114 47 88)(15 56 48 23)(16 116 45 86)(22 80 55 106)(24 78 53 108)(25 33 58 65)(26 100 59 126)(27 35 60 67)(28 98 57 128)(29 37 62 69)(31 39 64 71)(34 118 66 92)(36 120 68 90)(38 122 70 96)(40 124 72 94)(77 115 107 85)(79 113 105 87)(89 97 119 127)(91 99 117 125)
(1 91 75 117)(2 90 76 120)(3 89 73 119)(4 92 74 118)(5 114 102 88)(6 113 103 87)(7 116 104 86)(8 115 101 85)(9 58 42 25)(10 57 43 28)(11 60 44 27)(12 59 41 26)(13 62 46 29)(14 61 47 32)(15 64 48 31)(16 63 45 30)(17 65 50 33)(18 68 51 36)(19 67 52 35)(20 66 49 34)(21 69 54 37)(22 72 55 40)(23 71 56 39)(24 70 53 38)(77 123 107 93)(78 122 108 96)(79 121 105 95)(80 124 106 94)(81 127 111 97)(82 126 112 100)(83 125 109 99)(84 128 110 98)
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), (1,13,42,79)(2,14,43,80)(3,15,44,77)(4,16,41,78)(5,100,72,34)(6,97,69,35)(7,98,70,36)(8,99,71,33)(9,105,75,46)(10,106,76,47)(11,107,73,48)(12,108,74,45)(17,115,83,56)(18,116,84,53)(19,113,81,54)(20,114,82,55)(21,52,87,111)(22,49,88,112)(23,50,85,109)(24,51,86,110)(25,121,91,62)(26,122,92,63)(27,123,89,64)(28,124,90,61)(29,58,95,117)(30,59,96,118)(31,60,93,119)(32,57,94,120)(37,67,103,127)(38,68,104,128)(39,65,101,125)(40,66,102,126), (1,83,75,109)(2,18,76,51)(3,81,73,111)(4,20,74,49)(5,32,102,61)(6,95,103,121)(7,30,104,63)(8,93,101,123)(9,50,42,17)(10,110,43,84)(11,52,44,19)(12,112,41,82)(13,54,46,21)(14,114,47,88)(15,56,48,23)(16,116,45,86)(22,80,55,106)(24,78,53,108)(25,33,58,65)(26,100,59,126)(27,35,60,67)(28,98,57,128)(29,37,62,69)(31,39,64,71)(34,118,66,92)(36,120,68,90)(38,122,70,96)(40,124,72,94)(77,115,107,85)(79,113,105,87)(89,97,119,127)(91,99,117,125), (1,91,75,117)(2,90,76,120)(3,89,73,119)(4,92,74,118)(5,114,102,88)(6,113,103,87)(7,116,104,86)(8,115,101,85)(9,58,42,25)(10,57,43,28)(11,60,44,27)(12,59,41,26)(13,62,46,29)(14,61,47,32)(15,64,48,31)(16,63,45,30)(17,65,50,33)(18,68,51,36)(19,67,52,35)(20,66,49,34)(21,69,54,37)(22,72,55,40)(23,71,56,39)(24,70,53,38)(77,123,107,93)(78,122,108,96)(79,121,105,95)(80,124,106,94)(81,127,111,97)(82,126,112,100)(83,125,109,99)(84,128,110,98)>;
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), (1,13,42,79)(2,14,43,80)(3,15,44,77)(4,16,41,78)(5,100,72,34)(6,97,69,35)(7,98,70,36)(8,99,71,33)(9,105,75,46)(10,106,76,47)(11,107,73,48)(12,108,74,45)(17,115,83,56)(18,116,84,53)(19,113,81,54)(20,114,82,55)(21,52,87,111)(22,49,88,112)(23,50,85,109)(24,51,86,110)(25,121,91,62)(26,122,92,63)(27,123,89,64)(28,124,90,61)(29,58,95,117)(30,59,96,118)(31,60,93,119)(32,57,94,120)(37,67,103,127)(38,68,104,128)(39,65,101,125)(40,66,102,126), (1,83,75,109)(2,18,76,51)(3,81,73,111)(4,20,74,49)(5,32,102,61)(6,95,103,121)(7,30,104,63)(8,93,101,123)(9,50,42,17)(10,110,43,84)(11,52,44,19)(12,112,41,82)(13,54,46,21)(14,114,47,88)(15,56,48,23)(16,116,45,86)(22,80,55,106)(24,78,53,108)(25,33,58,65)(26,100,59,126)(27,35,60,67)(28,98,57,128)(29,37,62,69)(31,39,64,71)(34,118,66,92)(36,120,68,90)(38,122,70,96)(40,124,72,94)(77,115,107,85)(79,113,105,87)(89,97,119,127)(91,99,117,125), (1,91,75,117)(2,90,76,120)(3,89,73,119)(4,92,74,118)(5,114,102,88)(6,113,103,87)(7,116,104,86)(8,115,101,85)(9,58,42,25)(10,57,43,28)(11,60,44,27)(12,59,41,26)(13,62,46,29)(14,61,47,32)(15,64,48,31)(16,63,45,30)(17,65,50,33)(18,68,51,36)(19,67,52,35)(20,66,49,34)(21,69,54,37)(22,72,55,40)(23,71,56,39)(24,70,53,38)(77,123,107,93)(78,122,108,96)(79,121,105,95)(80,124,106,94)(81,127,111,97)(82,126,112,100)(83,125,109,99)(84,128,110,98) );
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)], [(1,13,42,79),(2,14,43,80),(3,15,44,77),(4,16,41,78),(5,100,72,34),(6,97,69,35),(7,98,70,36),(8,99,71,33),(9,105,75,46),(10,106,76,47),(11,107,73,48),(12,108,74,45),(17,115,83,56),(18,116,84,53),(19,113,81,54),(20,114,82,55),(21,52,87,111),(22,49,88,112),(23,50,85,109),(24,51,86,110),(25,121,91,62),(26,122,92,63),(27,123,89,64),(28,124,90,61),(29,58,95,117),(30,59,96,118),(31,60,93,119),(32,57,94,120),(37,67,103,127),(38,68,104,128),(39,65,101,125),(40,66,102,126)], [(1,83,75,109),(2,18,76,51),(3,81,73,111),(4,20,74,49),(5,32,102,61),(6,95,103,121),(7,30,104,63),(8,93,101,123),(9,50,42,17),(10,110,43,84),(11,52,44,19),(12,112,41,82),(13,54,46,21),(14,114,47,88),(15,56,48,23),(16,116,45,86),(22,80,55,106),(24,78,53,108),(25,33,58,65),(26,100,59,126),(27,35,60,67),(28,98,57,128),(29,37,62,69),(31,39,64,71),(34,118,66,92),(36,120,68,90),(38,122,70,96),(40,124,72,94),(77,115,107,85),(79,113,105,87),(89,97,119,127),(91,99,117,125)], [(1,91,75,117),(2,90,76,120),(3,89,73,119),(4,92,74,118),(5,114,102,88),(6,113,103,87),(7,116,104,86),(8,115,101,85),(9,58,42,25),(10,57,43,28),(11,60,44,27),(12,59,41,26),(13,62,46,29),(14,61,47,32),(15,64,48,31),(16,63,45,30),(17,65,50,33),(18,68,51,36),(19,67,52,35),(20,66,49,34),(21,69,54,37),(22,72,55,40),(23,71,56,39),(24,70,53,38),(77,123,107,93),(78,122,108,96),(79,121,105,95),(80,124,106,94),(81,127,111,97),(82,126,112,100),(83,125,109,99),(84,128,110,98)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊11Q8 | C4×C4⋊C4 | C42⋊8C4 | C42⋊5C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.78C23 | C23.81C23 | C23.83C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of C42⋊11Q8 ►in GL8(𝔽5)
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 1 | 3 | 3 | 1 |
| 0 | 0 | 0 | 0 | 4 | 4 | 2 | 4 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,2,4,1,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,3],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,4,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,3],[4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,3,0,1,4,0,0,0,0,2,0,3,4,0,0,0,0,1,3,3,2,0,0,0,0,4,3,1,4] >;
C42⋊11Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{11}Q_8 % in TeX
G:=Group("C4^2:11Q8"); // GroupNames label
G:=SmallGroup(128,1398);
// by ID
G=gap.SmallGroup(128,1398);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,352,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations