p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.180D4, C23.482C24, C22.1972- (1+4), C22.2642+ (1+4), C2.7(Q82), (C2×Q8)⋊12Q8, C42⋊9C4.34C2, C2.21(Q8⋊3Q8), C4.105(C22⋊Q8), (C2×C42).576C22, (C22×C4).112C23, C22.323(C22×D4), C22.117(C22×Q8), (C22×Q8).441C22, C23.81C23.18C2, C23.67C23.44C2, C2.C42.216C22, C23.65C23.59C2, C2.23(C22.53C24), C2.16(C22.31C24), (C2×C4×Q8).38C2, (C2×C4).59(C2×Q8), (C2×C4).363(C2×D4), C2.40(C2×C22⋊Q8), (C2×C4).897(C4○D4), (C2×C4⋊C4).328C22, C22.358(C2×C4○D4), SmallGroup(128,1314)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 242 conjugacy classes, 124 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×18], C22 [×3], C22 [×4], C2×C4 [×22], C2×C4 [×34], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×26], C22×C4, C22×C4 [×14], C2×Q8 [×8], C2×Q8 [×8], C2.C42 [×12], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×16], C4×Q8 [×8], C22×Q8 [×2], C42⋊9C4, C23.65C23 [×4], C23.67C23 [×4], C23.81C23 [×4], C2×C4×Q8 [×2], C42.180D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C22⋊Q8 [×8], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ (1+4), 2- (1+4), C2×C22⋊Q8 [×2], C22.31C24, Q8⋊3Q8 [×2], Q82, C22.53C24, C42.180D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2c-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 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)
(1 125 28 111)(2 128 25 110)(3 127 26 109)(4 126 27 112)(5 31 116 52)(6 30 113 51)(7 29 114 50)(8 32 115 49)(9 95 71 85)(10 94 72 88)(11 93 69 87)(12 96 70 86)(13 37 58 78)(14 40 59 77)(15 39 60 80)(16 38 57 79)(17 120 64 102)(18 119 61 101)(19 118 62 104)(20 117 63 103)(21 124 98 106)(22 123 99 105)(23 122 100 108)(24 121 97 107)(33 54 76 41)(34 53 73 44)(35 56 74 43)(36 55 75 42)(45 89 67 83)(46 92 68 82)(47 91 65 81)(48 90 66 84)
(1 86 44 103)(2 104 41 87)(3 88 42 101)(4 102 43 85)(5 67 37 98)(6 99 38 68)(7 65 39 100)(8 97 40 66)(9 110 17 76)(10 73 18 111)(11 112 19 74)(12 75 20 109)(13 108 52 81)(14 82 49 105)(15 106 50 83)(16 84 51 107)(21 116 45 78)(22 79 46 113)(23 114 47 80)(24 77 48 115)(25 118 54 93)(26 94 55 119)(27 120 56 95)(28 96 53 117)(29 89 60 124)(30 121 57 90)(31 91 58 122)(32 123 59 92)(33 71 128 64)(34 61 125 72)(35 69 126 62)(36 63 127 70)
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), (1,125,28,111)(2,128,25,110)(3,127,26,109)(4,126,27,112)(5,31,116,52)(6,30,113,51)(7,29,114,50)(8,32,115,49)(9,95,71,85)(10,94,72,88)(11,93,69,87)(12,96,70,86)(13,37,58,78)(14,40,59,77)(15,39,60,80)(16,38,57,79)(17,120,64,102)(18,119,61,101)(19,118,62,104)(20,117,63,103)(21,124,98,106)(22,123,99,105)(23,122,100,108)(24,121,97,107)(33,54,76,41)(34,53,73,44)(35,56,74,43)(36,55,75,42)(45,89,67,83)(46,92,68,82)(47,91,65,81)(48,90,66,84), (1,86,44,103)(2,104,41,87)(3,88,42,101)(4,102,43,85)(5,67,37,98)(6,99,38,68)(7,65,39,100)(8,97,40,66)(9,110,17,76)(10,73,18,111)(11,112,19,74)(12,75,20,109)(13,108,52,81)(14,82,49,105)(15,106,50,83)(16,84,51,107)(21,116,45,78)(22,79,46,113)(23,114,47,80)(24,77,48,115)(25,118,54,93)(26,94,55,119)(27,120,56,95)(28,96,53,117)(29,89,60,124)(30,121,57,90)(31,91,58,122)(32,123,59,92)(33,71,128,64)(34,61,125,72)(35,69,126,62)(36,63,127,70)>;
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), (1,125,28,111)(2,128,25,110)(3,127,26,109)(4,126,27,112)(5,31,116,52)(6,30,113,51)(7,29,114,50)(8,32,115,49)(9,95,71,85)(10,94,72,88)(11,93,69,87)(12,96,70,86)(13,37,58,78)(14,40,59,77)(15,39,60,80)(16,38,57,79)(17,120,64,102)(18,119,61,101)(19,118,62,104)(20,117,63,103)(21,124,98,106)(22,123,99,105)(23,122,100,108)(24,121,97,107)(33,54,76,41)(34,53,73,44)(35,56,74,43)(36,55,75,42)(45,89,67,83)(46,92,68,82)(47,91,65,81)(48,90,66,84), (1,86,44,103)(2,104,41,87)(3,88,42,101)(4,102,43,85)(5,67,37,98)(6,99,38,68)(7,65,39,100)(8,97,40,66)(9,110,17,76)(10,73,18,111)(11,112,19,74)(12,75,20,109)(13,108,52,81)(14,82,49,105)(15,106,50,83)(16,84,51,107)(21,116,45,78)(22,79,46,113)(23,114,47,80)(24,77,48,115)(25,118,54,93)(26,94,55,119)(27,120,56,95)(28,96,53,117)(29,89,60,124)(30,121,57,90)(31,91,58,122)(32,123,59,92)(33,71,128,64)(34,61,125,72)(35,69,126,62)(36,63,127,70) );
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)], [(1,125,28,111),(2,128,25,110),(3,127,26,109),(4,126,27,112),(5,31,116,52),(6,30,113,51),(7,29,114,50),(8,32,115,49),(9,95,71,85),(10,94,72,88),(11,93,69,87),(12,96,70,86),(13,37,58,78),(14,40,59,77),(15,39,60,80),(16,38,57,79),(17,120,64,102),(18,119,61,101),(19,118,62,104),(20,117,63,103),(21,124,98,106),(22,123,99,105),(23,122,100,108),(24,121,97,107),(33,54,76,41),(34,53,73,44),(35,56,74,43),(36,55,75,42),(45,89,67,83),(46,92,68,82),(47,91,65,81),(48,90,66,84)], [(1,86,44,103),(2,104,41,87),(3,88,42,101),(4,102,43,85),(5,67,37,98),(6,99,38,68),(7,65,39,100),(8,97,40,66),(9,110,17,76),(10,73,18,111),(11,112,19,74),(12,75,20,109),(13,108,52,81),(14,82,49,105),(15,106,50,83),(16,84,51,107),(21,116,45,78),(22,79,46,113),(23,114,47,80),(24,77,48,115),(25,118,54,93),(26,94,55,119),(27,120,56,95),(28,96,53,117),(29,89,60,124),(30,121,57,90),(31,91,58,122),(32,123,59,92),(33,71,128,64),(34,61,125,72),(35,69,126,62),(36,63,127,70)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
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 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42.180D4 | C42⋊9C4 | C23.65C23 | C23.67C23 | C23.81C23 | C2×C4×Q8 | C42 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 8 | 8 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_4^2._{180}D_4 % in TeX
G:=Group("C4^2.180D4"); // GroupNames label
G:=SmallGroup(128,1314);
// by ID
G=gap.SmallGroup(128,1314);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,352,675,80]);
// 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,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations