p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.34Q8, C23.302C24, C42⋊4C4.19C2, (C2×C42).30C22, C4.40(C42.C2), C22.62(C22×Q8), (C22×C4).502C23, C4○5(C23.81C23), C4○5(C23.83C23), C23.83C23.52C2, C2.C42.537C22, C23.81C23.56C2, C2.9(C23.37C23), C2.17(C23.36C23), (C4×C4⋊C4).47C2, (C2×C4).123(C2×Q8), C2.7(C2×C42.C2), (C2×C4).93(C4○D4), (C2×C4⋊C4).844C22, C22.182(C2×C4○D4), (C2×C4)○3(C23.81C23), SmallGroup(128,1134)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 308 in 200 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×20], C22, C22 [×6], C2×C4 [×18], C2×C4 [×36], C23, C42 [×4], C42 [×12], C4⋊C4 [×24], C22×C4, C22×C4 [×14], C2.C42 [×16], C2×C42, C2×C42 [×6], C2×C4⋊C4 [×12], C42⋊4C4, C4×C4⋊C4 [×6], C23.81C23 [×4], C23.83C23 [×4], C42.34Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×12], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×6], C2×C42.C2, C23.36C23 [×4], C23.37C23 [×2], C42.34Q8
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, 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 25 68 30)(2 26 65 31)(3 27 66 32)(4 28 67 29)(5 44 125 45)(6 41 126 46)(7 42 127 47)(8 43 128 48)(9 54 14 83)(10 55 15 84)(11 56 16 81)(12 53 13 82)(17 62 22 85)(18 63 23 86)(19 64 24 87)(20 61 21 88)(33 118 38 123)(34 119 39 124)(35 120 40 121)(36 117 37 122)(49 108 90 95)(50 105 91 96)(51 106 92 93)(52 107 89 94)(57 116 80 111)(58 113 77 112)(59 114 78 109)(60 115 79 110)(69 101 74 100)(70 102 75 97)(71 103 76 98)(72 104 73 99)
(1 125 117 75)(2 8 118 69)(3 127 119 73)(4 6 120 71)(5 122 70 68)(7 124 72 66)(9 24 51 80)(10 18 52 60)(11 22 49 78)(12 20 50 58)(13 21 91 77)(14 19 92 57)(15 23 89 79)(16 17 90 59)(25 45 37 97)(26 43 38 101)(27 47 39 99)(28 41 40 103)(29 46 35 98)(30 44 36 102)(31 48 33 100)(32 42 34 104)(53 61 105 113)(54 87 106 111)(55 63 107 115)(56 85 108 109)(62 95 114 81)(64 93 116 83)(65 128 123 74)(67 126 121 76)(82 88 96 112)(84 86 94 110)
(1 82 124 107)(2 54 121 95)(3 84 122 105)(4 56 123 93)(5 113 73 86)(6 109 74 64)(7 115 75 88)(8 111 76 62)(9 40 90 31)(10 36 91 27)(11 38 92 29)(12 34 89 25)(13 39 52 30)(14 35 49 26)(15 37 50 32)(16 33 51 28)(17 48 80 103)(18 44 77 99)(19 46 78 101)(20 42 79 97)(21 47 60 102)(22 43 57 98)(23 45 58 104)(24 41 59 100)(53 119 94 68)(55 117 96 66)(61 127 110 70)(63 125 112 72)(65 83 120 108)(67 81 118 106)(69 87 126 114)(71 85 128 116)
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), (1,25,68,30)(2,26,65,31)(3,27,66,32)(4,28,67,29)(5,44,125,45)(6,41,126,46)(7,42,127,47)(8,43,128,48)(9,54,14,83)(10,55,15,84)(11,56,16,81)(12,53,13,82)(17,62,22,85)(18,63,23,86)(19,64,24,87)(20,61,21,88)(33,118,38,123)(34,119,39,124)(35,120,40,121)(36,117,37,122)(49,108,90,95)(50,105,91,96)(51,106,92,93)(52,107,89,94)(57,116,80,111)(58,113,77,112)(59,114,78,109)(60,115,79,110)(69,101,74,100)(70,102,75,97)(71,103,76,98)(72,104,73,99), (1,125,117,75)(2,8,118,69)(3,127,119,73)(4,6,120,71)(5,122,70,68)(7,124,72,66)(9,24,51,80)(10,18,52,60)(11,22,49,78)(12,20,50,58)(13,21,91,77)(14,19,92,57)(15,23,89,79)(16,17,90,59)(25,45,37,97)(26,43,38,101)(27,47,39,99)(28,41,40,103)(29,46,35,98)(30,44,36,102)(31,48,33,100)(32,42,34,104)(53,61,105,113)(54,87,106,111)(55,63,107,115)(56,85,108,109)(62,95,114,81)(64,93,116,83)(65,128,123,74)(67,126,121,76)(82,88,96,112)(84,86,94,110), (1,82,124,107)(2,54,121,95)(3,84,122,105)(4,56,123,93)(5,113,73,86)(6,109,74,64)(7,115,75,88)(8,111,76,62)(9,40,90,31)(10,36,91,27)(11,38,92,29)(12,34,89,25)(13,39,52,30)(14,35,49,26)(15,37,50,32)(16,33,51,28)(17,48,80,103)(18,44,77,99)(19,46,78,101)(20,42,79,97)(21,47,60,102)(22,43,57,98)(23,45,58,104)(24,41,59,100)(53,119,94,68)(55,117,96,66)(61,127,110,70)(63,125,112,72)(65,83,120,108)(67,81,118,106)(69,87,126,114)(71,85,128,116)>;
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), (1,25,68,30)(2,26,65,31)(3,27,66,32)(4,28,67,29)(5,44,125,45)(6,41,126,46)(7,42,127,47)(8,43,128,48)(9,54,14,83)(10,55,15,84)(11,56,16,81)(12,53,13,82)(17,62,22,85)(18,63,23,86)(19,64,24,87)(20,61,21,88)(33,118,38,123)(34,119,39,124)(35,120,40,121)(36,117,37,122)(49,108,90,95)(50,105,91,96)(51,106,92,93)(52,107,89,94)(57,116,80,111)(58,113,77,112)(59,114,78,109)(60,115,79,110)(69,101,74,100)(70,102,75,97)(71,103,76,98)(72,104,73,99), (1,125,117,75)(2,8,118,69)(3,127,119,73)(4,6,120,71)(5,122,70,68)(7,124,72,66)(9,24,51,80)(10,18,52,60)(11,22,49,78)(12,20,50,58)(13,21,91,77)(14,19,92,57)(15,23,89,79)(16,17,90,59)(25,45,37,97)(26,43,38,101)(27,47,39,99)(28,41,40,103)(29,46,35,98)(30,44,36,102)(31,48,33,100)(32,42,34,104)(53,61,105,113)(54,87,106,111)(55,63,107,115)(56,85,108,109)(62,95,114,81)(64,93,116,83)(65,128,123,74)(67,126,121,76)(82,88,96,112)(84,86,94,110), (1,82,124,107)(2,54,121,95)(3,84,122,105)(4,56,123,93)(5,113,73,86)(6,109,74,64)(7,115,75,88)(8,111,76,62)(9,40,90,31)(10,36,91,27)(11,38,92,29)(12,34,89,25)(13,39,52,30)(14,35,49,26)(15,37,50,32)(16,33,51,28)(17,48,80,103)(18,44,77,99)(19,46,78,101)(20,42,79,97)(21,47,60,102)(22,43,57,98)(23,45,58,104)(24,41,59,100)(53,119,94,68)(55,117,96,66)(61,127,110,70)(63,125,112,72)(65,83,120,108)(67,81,118,106)(69,87,126,114)(71,85,128,116) );
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)], [(1,25,68,30),(2,26,65,31),(3,27,66,32),(4,28,67,29),(5,44,125,45),(6,41,126,46),(7,42,127,47),(8,43,128,48),(9,54,14,83),(10,55,15,84),(11,56,16,81),(12,53,13,82),(17,62,22,85),(18,63,23,86),(19,64,24,87),(20,61,21,88),(33,118,38,123),(34,119,39,124),(35,120,40,121),(36,117,37,122),(49,108,90,95),(50,105,91,96),(51,106,92,93),(52,107,89,94),(57,116,80,111),(58,113,77,112),(59,114,78,109),(60,115,79,110),(69,101,74,100),(70,102,75,97),(71,103,76,98),(72,104,73,99)], [(1,125,117,75),(2,8,118,69),(3,127,119,73),(4,6,120,71),(5,122,70,68),(7,124,72,66),(9,24,51,80),(10,18,52,60),(11,22,49,78),(12,20,50,58),(13,21,91,77),(14,19,92,57),(15,23,89,79),(16,17,90,59),(25,45,37,97),(26,43,38,101),(27,47,39,99),(28,41,40,103),(29,46,35,98),(30,44,36,102),(31,48,33,100),(32,42,34,104),(53,61,105,113),(54,87,106,111),(55,63,107,115),(56,85,108,109),(62,95,114,81),(64,93,116,83),(65,128,123,74),(67,126,121,76),(82,88,96,112),(84,86,94,110)], [(1,82,124,107),(2,54,121,95),(3,84,122,105),(4,56,123,93),(5,113,73,86),(6,109,74,64),(7,115,75,88),(8,111,76,62),(9,40,90,31),(10,36,91,27),(11,38,92,29),(12,34,89,25),(13,39,52,30),(14,35,49,26),(15,37,50,32),(16,33,51,28),(17,48,80,103),(18,44,77,99),(19,46,78,101),(20,42,79,97),(21,47,60,102),(22,43,57,98),(23,45,58,104),(24,41,59,100),(53,119,94,68),(55,117,96,66),(61,127,110,70),(63,125,112,72),(65,83,120,108),(67,81,118,106),(69,87,126,114),(71,85,128,116)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [0,3,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,1,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | C4○D4 |
| kernel | C42.34Q8 | C42⋊4C4 | C4×C4⋊C4 | C23.81C23 | C23.83C23 | C42 | C2×C4 |
| # reps | 1 | 1 | 6 | 4 | 4 | 4 | 24 |
In GAP, Magma, Sage, TeX
C_4^2._{34}Q_8 % in TeX
G:=Group("C4^2.34Q8"); // GroupNames label
G:=SmallGroup(128,1134);
// by ID
G=gap.SmallGroup(128,1134);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,184,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations