p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.191D4, C23.534C24, C22.2282- (1+4), C42⋊5C4.12C2, (C22×C4).144C23, (C2×C42).611C22, C22.359(C22×D4), (C22×Q8).450C22, C2.85(C22.19C24), C23.83C23.23C2, C23.81C23.26C2, C23.78C23.15C2, C2.C42.259C22, C23.63C23.37C2, C2.27(C22.35C24), C2.42(C23.38C23), (C2×C4×Q8).40C2, (C2×C4).393(C2×D4), (C2×C4).169(C4○D4), (C2×C4⋊C4).361C22, C22.406(C2×C4○D4), (C2×C42.C2).23C2, SmallGroup(128,1366)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 340 in 208 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×20], C22, C22 [×6], C2×C4 [×10], C2×C4 [×40], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×26], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×6], C2.C42 [×18], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4 [×10], C4×Q8 [×4], C42.C2 [×4], C22×Q8, C42⋊5C4, C23.63C23 [×4], C23.78C23 [×2], C23.81C23 [×2], C23.83C23 [×4], C2×C4×Q8, C2×C42.C2, C42.191D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2- (1+4) [×4], C22.19C24, C23.38C23 [×2], C22.35C24 [×4], C42.191D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=a2b, bd=db, 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 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 99 75 125)(2 36 76 68)(3 97 73 127)(4 34 74 66)(5 47 102 14)(6 105 103 79)(7 45 104 16)(8 107 101 77)(9 65 42 33)(10 128 43 98)(11 67 44 35)(12 126 41 100)(13 69 46 37)(15 71 48 39)(17 60 50 27)(18 118 51 92)(19 58 52 25)(20 120 49 90)(21 64 54 31)(22 122 55 96)(23 62 56 29)(24 124 53 94)(26 84 59 110)(28 82 57 112)(30 88 63 114)(32 86 61 116)(38 78 70 108)(40 80 72 106)(81 117 111 91)(83 119 109 89)(85 121 115 95)(87 123 113 93)
(1 119 3 117)(2 118 4 120)(5 114 7 116)(6 113 8 115)(9 27 11 25)(10 26 12 28)(13 31 15 29)(14 30 16 32)(17 35 19 33)(18 34 20 36)(21 39 23 37)(22 38 24 40)(41 57 43 59)(42 60 44 58)(45 61 47 63)(46 64 48 62)(49 68 51 66)(50 67 52 65)(53 72 55 70)(54 71 56 69)(73 91 75 89)(74 90 76 92)(77 95 79 93)(78 94 80 96)(81 99 83 97)(82 98 84 100)(85 103 87 101)(86 102 88 104)(105 123 107 121)(106 122 108 124)(109 127 111 125)(110 126 112 128)
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), (1,13,42,79)(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), (1,99,75,125)(2,36,76,68)(3,97,73,127)(4,34,74,66)(5,47,102,14)(6,105,103,79)(7,45,104,16)(8,107,101,77)(9,65,42,33)(10,128,43,98)(11,67,44,35)(12,126,41,100)(13,69,46,37)(15,71,48,39)(17,60,50,27)(18,118,51,92)(19,58,52,25)(20,120,49,90)(21,64,54,31)(22,122,55,96)(23,62,56,29)(24,124,53,94)(26,84,59,110)(28,82,57,112)(30,88,63,114)(32,86,61,116)(38,78,70,108)(40,80,72,106)(81,117,111,91)(83,119,109,89)(85,121,115,95)(87,123,113,93), (1,119,3,117)(2,118,4,120)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,57,43,59)(42,60,44,58)(45,61,47,63)(46,64,48,62)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,91,75,89)(74,90,76,92)(77,95,79,93)(78,94,80,96)(81,99,83,97)(82,98,84,100)(85,103,87,101)(86,102,88,104)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128)>;
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), (1,13,42,79)(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), (1,99,75,125)(2,36,76,68)(3,97,73,127)(4,34,74,66)(5,47,102,14)(6,105,103,79)(7,45,104,16)(8,107,101,77)(9,65,42,33)(10,128,43,98)(11,67,44,35)(12,126,41,100)(13,69,46,37)(15,71,48,39)(17,60,50,27)(18,118,51,92)(19,58,52,25)(20,120,49,90)(21,64,54,31)(22,122,55,96)(23,62,56,29)(24,124,53,94)(26,84,59,110)(28,82,57,112)(30,88,63,114)(32,86,61,116)(38,78,70,108)(40,80,72,106)(81,117,111,91)(83,119,109,89)(85,121,115,95)(87,123,113,93), (1,119,3,117)(2,118,4,120)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,57,43,59)(42,60,44,58)(45,61,47,63)(46,64,48,62)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,91,75,89)(74,90,76,92)(77,95,79,93)(78,94,80,96)(81,99,83,97)(82,98,84,100)(85,103,87,101)(86,102,88,104)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128) );
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)], [(1,13,42,79),(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)], [(1,99,75,125),(2,36,76,68),(3,97,73,127),(4,34,74,66),(5,47,102,14),(6,105,103,79),(7,45,104,16),(8,107,101,77),(9,65,42,33),(10,128,43,98),(11,67,44,35),(12,126,41,100),(13,69,46,37),(15,71,48,39),(17,60,50,27),(18,118,51,92),(19,58,52,25),(20,120,49,90),(21,64,54,31),(22,122,55,96),(23,62,56,29),(24,124,53,94),(26,84,59,110),(28,82,57,112),(30,88,63,114),(32,86,61,116),(38,78,70,108),(40,80,72,106),(81,117,111,91),(83,119,109,89),(85,121,115,95),(87,123,113,93)], [(1,119,3,117),(2,118,4,120),(5,114,7,116),(6,113,8,115),(9,27,11,25),(10,26,12,28),(13,31,15,29),(14,30,16,32),(17,35,19,33),(18,34,20,36),(21,39,23,37),(22,38,24,40),(41,57,43,59),(42,60,44,58),(45,61,47,63),(46,64,48,62),(49,68,51,66),(50,67,52,65),(53,72,55,70),(54,71,56,69),(73,91,75,89),(74,90,76,92),(77,95,79,93),(78,94,80,96),(81,99,83,97),(82,98,84,100),(85,103,87,101),(86,102,88,104),(105,123,107,121),(106,122,108,124),(109,127,111,125),(110,126,112,128)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 4 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 3 | 1 | 3 |
| 0 | 0 | 0 | 0 | 2 | 1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 2 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 4 | 3 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 2 | 1 |
| 0 | 0 | 0 | 0 | 4 | 1 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 4 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,2,1,0,0,0,0,0,3,1,3,0,0,0,0,2,1,2,0,0,0,0,0,1,3,0,2],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,2,4,0,0,0,0,0,1,1,0,4,0,0,0,0,4,0,1,3,0,0,0,0,0,4,4,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,4,2,0,0,0,0,0,1,1,1,0,0,0,0,4,2,4,0,0,0,0,0,1,1,0,4] >;
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 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2- (1+4) |
| kernel | C42.191D4 | C42⋊5C4 | C23.63C23 | C23.78C23 | C23.81C23 | C23.83C23 | C2×C4×Q8 | C2×C42.C2 | C42 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 4 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{191}D_4 % in TeX
G:=Group("C4^2.191D4"); // GroupNames label
G:=SmallGroup(128,1366);
// by ID
G=gap.SmallGroup(128,1366);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*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