p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.38Q8, C23.506C24, C22.2092- (1+4), C22.2862+ (1+4), C42⋊5C4.11C2, C42⋊4C4.23C2, (C22×C4).125C23, (C2×C42).593C22, C22.127(C22×Q8), C2.C42.551C22, C23.65C23.65C2, C23.63C23.35C2, C23.81C23.24C2, C2.76(C22.46C24), C2.38(C23.37C23), C2.17(C23.41C23), C2.80(C22.47C24), (C4×C4⋊C4).77C2, (C2×C4).128(C2×Q8), (C2×C4).164(C4○D4), (C2×C4⋊C4).345C22, C22.382(C2×C4○D4), SmallGroup(128,1338)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 308 in 190 conjugacy classes, 100 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×14], C2×C4 [×38], C23, C42 [×4], C42 [×6], C4⋊C4 [×24], C22×C4 [×3], C22×C4 [×12], C2.C42 [×4], C2.C42 [×12], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×14], C42⋊4C4, C4×C4⋊C4, C42⋊5C4, C23.63C23 [×4], C23.65C23 [×4], C23.81C23 [×4], C42.38Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C23.37C23 [×2], C23.41C23, C22.46C24 [×2], C22.47C24 [×2], C42.38Q8
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c-1 >
(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 16 59 27)(2 13 60 28)(3 14 57 25)(4 15 58 26)(5 35 111 81)(6 36 112 82)(7 33 109 83)(8 34 110 84)(9 56 32 41)(10 53 29 42)(11 54 30 43)(12 55 31 44)(17 64 68 49)(18 61 65 50)(19 62 66 51)(20 63 67 52)(21 100 72 45)(22 97 69 46)(23 98 70 47)(24 99 71 48)(37 113 87 126)(38 114 88 127)(39 115 85 128)(40 116 86 125)(73 124 94 103)(74 121 95 104)(75 122 96 101)(76 123 93 102)(77 120 90 107)(78 117 91 108)(79 118 92 105)(80 119 89 106)
(1 61 53 48)(2 51 54 100)(3 63 55 46)(4 49 56 98)(5 118 113 103)(6 106 114 121)(7 120 115 101)(8 108 116 123)(9 23 26 68)(10 71 27 18)(11 21 28 66)(12 69 25 20)(13 19 30 72)(14 67 31 22)(15 17 32 70)(16 65 29 24)(33 90 85 75)(34 78 86 93)(35 92 87 73)(36 80 88 95)(37 94 81 79)(38 74 82 89)(39 96 83 77)(40 76 84 91)(41 47 58 64)(42 99 59 50)(43 45 60 62)(44 97 57 52)(102 110 117 125)(104 112 119 127)(105 126 124 111)(107 128 122 109)
(1 117 55 104)(2 120 56 103)(3 119 53 102)(4 118 54 101)(5 100 115 49)(6 99 116 52)(7 98 113 51)(8 97 114 50)(9 96 28 79)(10 95 25 78)(11 94 26 77)(12 93 27 80)(13 92 32 75)(14 91 29 74)(15 90 30 73)(16 89 31 76)(17 33 72 87)(18 36 69 86)(19 35 70 85)(20 34 71 88)(21 37 68 83)(22 40 65 82)(23 39 66 81)(24 38 67 84)(41 124 60 107)(42 123 57 106)(43 122 58 105)(44 121 59 108)(45 128 64 111)(46 127 61 110)(47 126 62 109)(48 125 63 112)
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), (1,16,59,27)(2,13,60,28)(3,14,57,25)(4,15,58,26)(5,35,111,81)(6,36,112,82)(7,33,109,83)(8,34,110,84)(9,56,32,41)(10,53,29,42)(11,54,30,43)(12,55,31,44)(17,64,68,49)(18,61,65,50)(19,62,66,51)(20,63,67,52)(21,100,72,45)(22,97,69,46)(23,98,70,47)(24,99,71,48)(37,113,87,126)(38,114,88,127)(39,115,85,128)(40,116,86,125)(73,124,94,103)(74,121,95,104)(75,122,96,101)(76,123,93,102)(77,120,90,107)(78,117,91,108)(79,118,92,105)(80,119,89,106), (1,61,53,48)(2,51,54,100)(3,63,55,46)(4,49,56,98)(5,118,113,103)(6,106,114,121)(7,120,115,101)(8,108,116,123)(9,23,26,68)(10,71,27,18)(11,21,28,66)(12,69,25,20)(13,19,30,72)(14,67,31,22)(15,17,32,70)(16,65,29,24)(33,90,85,75)(34,78,86,93)(35,92,87,73)(36,80,88,95)(37,94,81,79)(38,74,82,89)(39,96,83,77)(40,76,84,91)(41,47,58,64)(42,99,59,50)(43,45,60,62)(44,97,57,52)(102,110,117,125)(104,112,119,127)(105,126,124,111)(107,128,122,109), (1,117,55,104)(2,120,56,103)(3,119,53,102)(4,118,54,101)(5,100,115,49)(6,99,116,52)(7,98,113,51)(8,97,114,50)(9,96,28,79)(10,95,25,78)(11,94,26,77)(12,93,27,80)(13,92,32,75)(14,91,29,74)(15,90,30,73)(16,89,31,76)(17,33,72,87)(18,36,69,86)(19,35,70,85)(20,34,71,88)(21,37,68,83)(22,40,65,82)(23,39,66,81)(24,38,67,84)(41,124,60,107)(42,123,57,106)(43,122,58,105)(44,121,59,108)(45,128,64,111)(46,127,61,110)(47,126,62,109)(48,125,63,112)>;
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), (1,16,59,27)(2,13,60,28)(3,14,57,25)(4,15,58,26)(5,35,111,81)(6,36,112,82)(7,33,109,83)(8,34,110,84)(9,56,32,41)(10,53,29,42)(11,54,30,43)(12,55,31,44)(17,64,68,49)(18,61,65,50)(19,62,66,51)(20,63,67,52)(21,100,72,45)(22,97,69,46)(23,98,70,47)(24,99,71,48)(37,113,87,126)(38,114,88,127)(39,115,85,128)(40,116,86,125)(73,124,94,103)(74,121,95,104)(75,122,96,101)(76,123,93,102)(77,120,90,107)(78,117,91,108)(79,118,92,105)(80,119,89,106), (1,61,53,48)(2,51,54,100)(3,63,55,46)(4,49,56,98)(5,118,113,103)(6,106,114,121)(7,120,115,101)(8,108,116,123)(9,23,26,68)(10,71,27,18)(11,21,28,66)(12,69,25,20)(13,19,30,72)(14,67,31,22)(15,17,32,70)(16,65,29,24)(33,90,85,75)(34,78,86,93)(35,92,87,73)(36,80,88,95)(37,94,81,79)(38,74,82,89)(39,96,83,77)(40,76,84,91)(41,47,58,64)(42,99,59,50)(43,45,60,62)(44,97,57,52)(102,110,117,125)(104,112,119,127)(105,126,124,111)(107,128,122,109), (1,117,55,104)(2,120,56,103)(3,119,53,102)(4,118,54,101)(5,100,115,49)(6,99,116,52)(7,98,113,51)(8,97,114,50)(9,96,28,79)(10,95,25,78)(11,94,26,77)(12,93,27,80)(13,92,32,75)(14,91,29,74)(15,90,30,73)(16,89,31,76)(17,33,72,87)(18,36,69,86)(19,35,70,85)(20,34,71,88)(21,37,68,83)(22,40,65,82)(23,39,66,81)(24,38,67,84)(41,124,60,107)(42,123,57,106)(43,122,58,105)(44,121,59,108)(45,128,64,111)(46,127,61,110)(47,126,62,109)(48,125,63,112) );
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)], [(1,16,59,27),(2,13,60,28),(3,14,57,25),(4,15,58,26),(5,35,111,81),(6,36,112,82),(7,33,109,83),(8,34,110,84),(9,56,32,41),(10,53,29,42),(11,54,30,43),(12,55,31,44),(17,64,68,49),(18,61,65,50),(19,62,66,51),(20,63,67,52),(21,100,72,45),(22,97,69,46),(23,98,70,47),(24,99,71,48),(37,113,87,126),(38,114,88,127),(39,115,85,128),(40,116,86,125),(73,124,94,103),(74,121,95,104),(75,122,96,101),(76,123,93,102),(77,120,90,107),(78,117,91,108),(79,118,92,105),(80,119,89,106)], [(1,61,53,48),(2,51,54,100),(3,63,55,46),(4,49,56,98),(5,118,113,103),(6,106,114,121),(7,120,115,101),(8,108,116,123),(9,23,26,68),(10,71,27,18),(11,21,28,66),(12,69,25,20),(13,19,30,72),(14,67,31,22),(15,17,32,70),(16,65,29,24),(33,90,85,75),(34,78,86,93),(35,92,87,73),(36,80,88,95),(37,94,81,79),(38,74,82,89),(39,96,83,77),(40,76,84,91),(41,47,58,64),(42,99,59,50),(43,45,60,62),(44,97,57,52),(102,110,117,125),(104,112,119,127),(105,126,124,111),(107,128,122,109)], [(1,117,55,104),(2,120,56,103),(3,119,53,102),(4,118,54,101),(5,100,115,49),(6,99,116,52),(7,98,113,51),(8,97,114,50),(9,96,28,79),(10,95,25,78),(11,94,26,77),(12,93,27,80),(13,92,32,75),(14,91,29,74),(15,90,30,73),(16,89,31,76),(17,33,72,87),(18,36,69,86),(19,35,70,85),(20,34,71,88),(21,37,68,83),(22,40,65,82),(23,39,66,81),(24,38,67,84),(41,124,60,107),(42,123,57,106),(43,122,58,105),(44,121,59,108),(45,128,64,111),(46,127,61,110),(47,126,62,109),(48,125,63,112)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 3 | 3 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 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 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 2 | 0 |
G:=sub<GL(6,GF(5))| [3,4,0,0,0,0,3,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[2,0,0,0,0,0,2,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[1,3,0,0,0,0,1,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0] >;
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 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42.38Q8 | C42⋊4C4 | C4×C4⋊C4 | C42⋊5C4 | C23.63C23 | C23.65C23 | C23.81C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 16 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_4^2._{38}Q_8 % in TeX
G:=Group("C4^2.38Q8"); // GroupNames label
G:=SmallGroup(128,1338);
// by ID
G=gap.SmallGroup(128,1338);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,184,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations