p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊7Q8, C23.451C24, C22.2362+ 1+4, C22.1822- 1+4, C2.5Q82, C2.30D42, C4⋊C4⋊19Q8, C4⋊2(C4⋊Q8), C2.24(D4×Q8), C4⋊C4.234D4, C42⋊9C4.28C2, (C22×C4).840C23, (C2×C42).556C22, C22.302(C22×D4), C22.101(C22×Q8), (C22×Q8).134C22, C23.78C23.7C2, C23.65C23.53C2, C2.C42.188C22, C2.12(C23.41C23), (C4×C4⋊C4).64C2, C2.14(C2×C4⋊Q8), (C2×C4).77(C2×D4), (C2×C4⋊Q8).33C2, (C2×C4).51(C2×Q8), (C2×C4⋊C4).871C22, SmallGroup(128,1283)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊7Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 468 in 278 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C4×C4⋊C4, C42⋊9C4, C23.65C23, C23.78C23, C2×C4⋊Q8, C42⋊7Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C22×D4, C22×Q8, 2+ 1+4, 2- 1+4, C2×C4⋊Q8, C23.41C23, D42, D4×Q8, Q82, C42⋊7Q8
►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 15 44 50)(2 16 41 51)(3 13 42 52)(4 14 43 49)(5 36 37 127)(6 33 38 128)(7 34 39 125)(8 35 40 126)(9 22 17 46)(10 23 18 47)(11 24 19 48)(12 21 20 45)(25 57 54 30)(26 58 55 31)(27 59 56 32)(28 60 53 29)(61 65 72 100)(62 66 69 97)(63 67 70 98)(64 68 71 99)(73 80 111 114)(74 77 112 115)(75 78 109 116)(76 79 110 113)(81 101 108 88)(82 102 105 85)(83 103 106 86)(84 104 107 87)(89 117 124 96)(90 118 121 93)(91 119 122 94)(92 120 123 95)
(1 70 26 10)(2 62 27 17)(3 72 28 12)(4 64 25 19)(5 106 114 122)(6 82 115 90)(7 108 116 124)(8 84 113 92)(9 41 69 56)(11 43 71 54)(13 100 60 21)(14 68 57 48)(15 98 58 23)(16 66 59 46)(18 44 63 55)(20 42 61 53)(22 51 97 32)(24 49 99 30)(29 45 52 65)(31 47 50 67)(33 102 74 118)(34 88 75 96)(35 104 76 120)(36 86 73 94)(37 83 80 91)(38 105 77 121)(39 81 78 89)(40 107 79 123)(85 112 93 128)(87 110 95 126)(101 109 117 125)(103 111 119 127)
(1 119 26 103)(2 95 27 87)(3 117 28 101)(4 93 25 85)(5 67 114 47)(6 99 115 24)(7 65 116 45)(8 97 113 22)(9 35 69 76)(10 127 70 111)(11 33 71 74)(12 125 72 109)(13 89 60 81)(14 121 57 105)(15 91 58 83)(16 123 59 107)(17 126 62 110)(18 36 63 73)(19 128 64 112)(20 34 61 75)(21 39 100 78)(23 37 98 80)(29 108 52 124)(30 82 49 90)(31 106 50 122)(32 84 51 92)(38 68 77 48)(40 66 79 46)(41 120 56 104)(42 96 53 88)(43 118 54 102)(44 94 55 86)
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), (1,15,44,50)(2,16,41,51)(3,13,42,52)(4,14,43,49)(5,36,37,127)(6,33,38,128)(7,34,39,125)(8,35,40,126)(9,22,17,46)(10,23,18,47)(11,24,19,48)(12,21,20,45)(25,57,54,30)(26,58,55,31)(27,59,56,32)(28,60,53,29)(61,65,72,100)(62,66,69,97)(63,67,70,98)(64,68,71,99)(73,80,111,114)(74,77,112,115)(75,78,109,116)(76,79,110,113)(81,101,108,88)(82,102,105,85)(83,103,106,86)(84,104,107,87)(89,117,124,96)(90,118,121,93)(91,119,122,94)(92,120,123,95), (1,70,26,10)(2,62,27,17)(3,72,28,12)(4,64,25,19)(5,106,114,122)(6,82,115,90)(7,108,116,124)(8,84,113,92)(9,41,69,56)(11,43,71,54)(13,100,60,21)(14,68,57,48)(15,98,58,23)(16,66,59,46)(18,44,63,55)(20,42,61,53)(22,51,97,32)(24,49,99,30)(29,45,52,65)(31,47,50,67)(33,102,74,118)(34,88,75,96)(35,104,76,120)(36,86,73,94)(37,83,80,91)(38,105,77,121)(39,81,78,89)(40,107,79,123)(85,112,93,128)(87,110,95,126)(101,109,117,125)(103,111,119,127), (1,119,26,103)(2,95,27,87)(3,117,28,101)(4,93,25,85)(5,67,114,47)(6,99,115,24)(7,65,116,45)(8,97,113,22)(9,35,69,76)(10,127,70,111)(11,33,71,74)(12,125,72,109)(13,89,60,81)(14,121,57,105)(15,91,58,83)(16,123,59,107)(17,126,62,110)(18,36,63,73)(19,128,64,112)(20,34,61,75)(21,39,100,78)(23,37,98,80)(29,108,52,124)(30,82,49,90)(31,106,50,122)(32,84,51,92)(38,68,77,48)(40,66,79,46)(41,120,56,104)(42,96,53,88)(43,118,54,102)(44,94,55,86)>;
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), (1,15,44,50)(2,16,41,51)(3,13,42,52)(4,14,43,49)(5,36,37,127)(6,33,38,128)(7,34,39,125)(8,35,40,126)(9,22,17,46)(10,23,18,47)(11,24,19,48)(12,21,20,45)(25,57,54,30)(26,58,55,31)(27,59,56,32)(28,60,53,29)(61,65,72,100)(62,66,69,97)(63,67,70,98)(64,68,71,99)(73,80,111,114)(74,77,112,115)(75,78,109,116)(76,79,110,113)(81,101,108,88)(82,102,105,85)(83,103,106,86)(84,104,107,87)(89,117,124,96)(90,118,121,93)(91,119,122,94)(92,120,123,95), (1,70,26,10)(2,62,27,17)(3,72,28,12)(4,64,25,19)(5,106,114,122)(6,82,115,90)(7,108,116,124)(8,84,113,92)(9,41,69,56)(11,43,71,54)(13,100,60,21)(14,68,57,48)(15,98,58,23)(16,66,59,46)(18,44,63,55)(20,42,61,53)(22,51,97,32)(24,49,99,30)(29,45,52,65)(31,47,50,67)(33,102,74,118)(34,88,75,96)(35,104,76,120)(36,86,73,94)(37,83,80,91)(38,105,77,121)(39,81,78,89)(40,107,79,123)(85,112,93,128)(87,110,95,126)(101,109,117,125)(103,111,119,127), (1,119,26,103)(2,95,27,87)(3,117,28,101)(4,93,25,85)(5,67,114,47)(6,99,115,24)(7,65,116,45)(8,97,113,22)(9,35,69,76)(10,127,70,111)(11,33,71,74)(12,125,72,109)(13,89,60,81)(14,121,57,105)(15,91,58,83)(16,123,59,107)(17,126,62,110)(18,36,63,73)(19,128,64,112)(20,34,61,75)(21,39,100,78)(23,37,98,80)(29,108,52,124)(30,82,49,90)(31,106,50,122)(32,84,51,92)(38,68,77,48)(40,66,79,46)(41,120,56,104)(42,96,53,88)(43,118,54,102)(44,94,55,86) );
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)], [(1,15,44,50),(2,16,41,51),(3,13,42,52),(4,14,43,49),(5,36,37,127),(6,33,38,128),(7,34,39,125),(8,35,40,126),(9,22,17,46),(10,23,18,47),(11,24,19,48),(12,21,20,45),(25,57,54,30),(26,58,55,31),(27,59,56,32),(28,60,53,29),(61,65,72,100),(62,66,69,97),(63,67,70,98),(64,68,71,99),(73,80,111,114),(74,77,112,115),(75,78,109,116),(76,79,110,113),(81,101,108,88),(82,102,105,85),(83,103,106,86),(84,104,107,87),(89,117,124,96),(90,118,121,93),(91,119,122,94),(92,120,123,95)], [(1,70,26,10),(2,62,27,17),(3,72,28,12),(4,64,25,19),(5,106,114,122),(6,82,115,90),(7,108,116,124),(8,84,113,92),(9,41,69,56),(11,43,71,54),(13,100,60,21),(14,68,57,48),(15,98,58,23),(16,66,59,46),(18,44,63,55),(20,42,61,53),(22,51,97,32),(24,49,99,30),(29,45,52,65),(31,47,50,67),(33,102,74,118),(34,88,75,96),(35,104,76,120),(36,86,73,94),(37,83,80,91),(38,105,77,121),(39,81,78,89),(40,107,79,123),(85,112,93,128),(87,110,95,126),(101,109,117,125),(103,111,119,127)], [(1,119,26,103),(2,95,27,87),(3,117,28,101),(4,93,25,85),(5,67,114,47),(6,99,115,24),(7,65,116,45),(8,97,113,22),(9,35,69,76),(10,127,70,111),(11,33,71,74),(12,125,72,109),(13,89,60,81),(14,121,57,105),(15,91,58,83),(16,123,59,107),(17,126,62,110),(18,36,63,73),(19,128,64,112),(20,34,61,75),(21,39,100,78),(23,37,98,80),(29,108,52,124),(30,82,49,90),(31,106,50,122),(32,84,51,92),(38,68,77,48),(40,66,79,46),(41,120,56,104),(42,96,53,88),(43,118,54,102),(44,94,55,86)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC | 4AD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | Q8 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊7Q8 | C4×C4⋊C4 | C42⋊9C4 | C23.65C23 | C23.78C23 | C2×C4⋊Q8 | C42 | C4⋊C4 | C4⋊C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 |
Matrix representation of C42⋊7Q8 ►in GL6(𝔽5)
| 4 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 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 | 4 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(5))| [4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,0,0,0,0,0,4,2,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0] >;
C42⋊7Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7Q_8
% in TeX
G:=Group("C4^2:7Q8"); // GroupNames label
G:=SmallGroup(128,1283);
// by ID
G=gap.SmallGroup(128,1283);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,184,675,80]);
// 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^-1*b^2,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations