p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.179D4, C23.481C24, C22.1962- (1+4), (C2×Q8)⋊11Q8, C42⋊8C4.37C2, C2.20(Q8⋊3Q8), C4.104(C22⋊Q8), (C2×C42).575C22, (C22×C4).111C23, C22.322(C22×D4), C22.116(C22×Q8), (C22×Q8).440C22, C2.66(C22.19C24), C23.78C23.9C2, C2.C42.215C22, C23.63C23.29C2, C23.65C23.58C2, C23.67C23.43C2, C23.83C23.17C2, C2.45(C22.50C24), C2.27(C23.38C23), (C2×C4×Q8).37C2, (C2×C4).58(C2×Q8), C2.39(C2×C22⋊Q8), (C2×C4).1197(C2×D4), (C2×C4).156(C4○D4), (C2×C4⋊C4).327C22, C22.357(C2×C4○D4), SmallGroup(128,1313)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 234 conjugacy classes, 112 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×18], C2×C4 [×36], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×24], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×10], C2.C42 [×16], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C4×Q8 [×8], C22×Q8 [×2], C42⋊8C4, C23.63C23 [×4], C23.65C23 [×2], C23.67C23 [×2], C23.78C23 [×2], C23.83C23 [×2], C2×C4×Q8 [×2], C42.179D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2- (1+4) [×2], C2×C22⋊Q8, C22.19C24, C23.38C23, C22.50C24 [×2], Q8⋊3Q8 [×2], C42.179D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2b2c-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 15 21 11)(2 16 22 12)(3 13 23 9)(4 14 24 10)(5 56 38 26)(6 53 39 27)(7 54 40 28)(8 55 37 25)(17 30 50 60)(18 31 51 57)(19 32 52 58)(20 29 49 59)(33 74 125 108)(34 75 126 105)(35 76 127 106)(36 73 128 107)(41 72 48 62)(42 69 45 63)(43 70 46 64)(44 71 47 61)(65 120 99 94)(66 117 100 95)(67 118 97 96)(68 119 98 93)(77 82 103 116)(78 83 104 113)(79 84 101 114)(80 81 102 115)(85 90 111 124)(86 91 112 121)(87 92 109 122)(88 89 110 123)
(1 74 6 89)(2 107 7 122)(3 76 8 91)(4 105 5 124)(9 127 25 112)(10 34 26 85)(11 125 27 110)(12 36 28 87)(13 35 55 86)(14 126 56 111)(15 33 53 88)(16 128 54 109)(17 118 62 103)(18 95 63 80)(19 120 64 101)(20 93 61 78)(21 108 39 123)(22 73 40 92)(23 106 37 121)(24 75 38 90)(29 98 44 113)(30 67 41 82)(31 100 42 115)(32 65 43 84)(45 81 57 66)(46 114 58 99)(47 83 59 68)(48 116 60 97)(49 119 71 104)(50 96 72 77)(51 117 69 102)(52 94 70 79)
(1 115 21 81)(2 82 22 116)(3 113 23 83)(4 84 24 114)(5 65 38 99)(6 100 39 66)(7 67 40 97)(8 98 37 68)(9 78 13 104)(10 101 14 79)(11 80 15 102)(12 103 16 77)(17 85 50 111)(18 112 51 86)(19 87 52 109)(20 110 49 88)(25 93 55 119)(26 120 56 94)(27 95 53 117)(28 118 54 96)(29 89 59 123)(30 124 60 90)(31 91 57 121)(32 122 58 92)(33 61 125 71)(34 72 126 62)(35 63 127 69)(36 70 128 64)(41 105 48 75)(42 76 45 106)(43 107 46 73)(44 74 47 108)
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,74,125,108)(34,75,126,105)(35,76,127,106)(36,73,128,107)(41,72,48,62)(42,69,45,63)(43,70,46,64)(44,71,47,61)(65,120,99,94)(66,117,100,95)(67,118,97,96)(68,119,98,93)(77,82,103,116)(78,83,104,113)(79,84,101,114)(80,81,102,115)(85,90,111,124)(86,91,112,121)(87,92,109,122)(88,89,110,123), (1,74,6,89)(2,107,7,122)(3,76,8,91)(4,105,5,124)(9,127,25,112)(10,34,26,85)(11,125,27,110)(12,36,28,87)(13,35,55,86)(14,126,56,111)(15,33,53,88)(16,128,54,109)(17,118,62,103)(18,95,63,80)(19,120,64,101)(20,93,61,78)(21,108,39,123)(22,73,40,92)(23,106,37,121)(24,75,38,90)(29,98,44,113)(30,67,41,82)(31,100,42,115)(32,65,43,84)(45,81,57,66)(46,114,58,99)(47,83,59,68)(48,116,60,97)(49,119,71,104)(50,96,72,77)(51,117,69,102)(52,94,70,79), (1,115,21,81)(2,82,22,116)(3,113,23,83)(4,84,24,114)(5,65,38,99)(6,100,39,66)(7,67,40,97)(8,98,37,68)(9,78,13,104)(10,101,14,79)(11,80,15,102)(12,103,16,77)(17,85,50,111)(18,112,51,86)(19,87,52,109)(20,110,49,88)(25,93,55,119)(26,120,56,94)(27,95,53,117)(28,118,54,96)(29,89,59,123)(30,124,60,90)(31,91,57,121)(32,122,58,92)(33,61,125,71)(34,72,126,62)(35,63,127,69)(36,70,128,64)(41,105,48,75)(42,76,45,106)(43,107,46,73)(44,74,47,108)>;
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,74,125,108)(34,75,126,105)(35,76,127,106)(36,73,128,107)(41,72,48,62)(42,69,45,63)(43,70,46,64)(44,71,47,61)(65,120,99,94)(66,117,100,95)(67,118,97,96)(68,119,98,93)(77,82,103,116)(78,83,104,113)(79,84,101,114)(80,81,102,115)(85,90,111,124)(86,91,112,121)(87,92,109,122)(88,89,110,123), (1,74,6,89)(2,107,7,122)(3,76,8,91)(4,105,5,124)(9,127,25,112)(10,34,26,85)(11,125,27,110)(12,36,28,87)(13,35,55,86)(14,126,56,111)(15,33,53,88)(16,128,54,109)(17,118,62,103)(18,95,63,80)(19,120,64,101)(20,93,61,78)(21,108,39,123)(22,73,40,92)(23,106,37,121)(24,75,38,90)(29,98,44,113)(30,67,41,82)(31,100,42,115)(32,65,43,84)(45,81,57,66)(46,114,58,99)(47,83,59,68)(48,116,60,97)(49,119,71,104)(50,96,72,77)(51,117,69,102)(52,94,70,79), (1,115,21,81)(2,82,22,116)(3,113,23,83)(4,84,24,114)(5,65,38,99)(6,100,39,66)(7,67,40,97)(8,98,37,68)(9,78,13,104)(10,101,14,79)(11,80,15,102)(12,103,16,77)(17,85,50,111)(18,112,51,86)(19,87,52,109)(20,110,49,88)(25,93,55,119)(26,120,56,94)(27,95,53,117)(28,118,54,96)(29,89,59,123)(30,124,60,90)(31,91,57,121)(32,122,58,92)(33,61,125,71)(34,72,126,62)(35,63,127,69)(36,70,128,64)(41,105,48,75)(42,76,45,106)(43,107,46,73)(44,74,47,108) );
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)], [(1,15,21,11),(2,16,22,12),(3,13,23,9),(4,14,24,10),(5,56,38,26),(6,53,39,27),(7,54,40,28),(8,55,37,25),(17,30,50,60),(18,31,51,57),(19,32,52,58),(20,29,49,59),(33,74,125,108),(34,75,126,105),(35,76,127,106),(36,73,128,107),(41,72,48,62),(42,69,45,63),(43,70,46,64),(44,71,47,61),(65,120,99,94),(66,117,100,95),(67,118,97,96),(68,119,98,93),(77,82,103,116),(78,83,104,113),(79,84,101,114),(80,81,102,115),(85,90,111,124),(86,91,112,121),(87,92,109,122),(88,89,110,123)], [(1,74,6,89),(2,107,7,122),(3,76,8,91),(4,105,5,124),(9,127,25,112),(10,34,26,85),(11,125,27,110),(12,36,28,87),(13,35,55,86),(14,126,56,111),(15,33,53,88),(16,128,54,109),(17,118,62,103),(18,95,63,80),(19,120,64,101),(20,93,61,78),(21,108,39,123),(22,73,40,92),(23,106,37,121),(24,75,38,90),(29,98,44,113),(30,67,41,82),(31,100,42,115),(32,65,43,84),(45,81,57,66),(46,114,58,99),(47,83,59,68),(48,116,60,97),(49,119,71,104),(50,96,72,77),(51,117,69,102),(52,94,70,79)], [(1,115,21,81),(2,82,22,116),(3,113,23,83),(4,84,24,114),(5,65,38,99),(6,100,39,66),(7,67,40,97),(8,98,37,68),(9,78,13,104),(10,101,14,79),(11,80,15,102),(12,103,16,77),(17,85,50,111),(18,112,51,86),(19,87,52,109),(20,110,49,88),(25,93,55,119),(26,120,56,94),(27,95,53,117),(28,118,54,96),(29,89,59,123),(30,124,60,90),(31,91,57,121),(32,122,58,92),(33,61,125,71),(34,72,126,62),(35,63,127,69),(36,70,128,64),(41,105,48,75),(42,76,45,106),(43,107,46,73),(44,74,47,108)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [0,2,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,3,2,0,0,0,0,0,2],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
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 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2- (1+4) |
| kernel | C42.179D4 | C42⋊8C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.78C23 | C23.83C23 | C2×C4×Q8 | C42 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{179}D_4 % in TeX
G:=Group("C4^2.179D4"); // GroupNames label
G:=SmallGroup(128,1313);
// by ID
G=gap.SmallGroup(128,1313);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,352,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations