p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊8Q8, C23.505C24, C22.2852+ (1+4), C22.2082- (1+4), C42⋊4C4.22C2, C42⋊5C4.10C2, (C2×C42).592C22, (C22×C4).124C23, C22.126(C22×Q8), (C22×Q8).147C22, C2.70(C22.45C24), C23.83C23.21C2, C23.78C23.13C2, C23.63C23.34C2, C2.C42.235C22, C23.67C23.47C2, C2.16(C23.41C23), C2.37(C23.37C23), C2.53(C22.50C24), (C4×C4⋊C4).76C2, (C2×C4).127(C2×Q8), (C2×C4).163(C4○D4), (C2×C4⋊C4).344C22, C22.381(C2×C4○D4), SmallGroup(128,1337)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 340 in 198 conjugacy classes, 100 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×14], C2×C4 [×38], Q8 [×8], C23, C42 [×4], C42 [×6], C4⋊C4 [×14], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×10], C2.C42 [×4], C2.C42 [×16], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×8], C22×Q8 [×2], C42⋊4C4, C4×C4⋊C4, C42⋊5C4, C23.63C23 [×4], C23.67C23 [×4], C23.78C23 [×2], C23.83C23 [×2], C42⋊8Q8
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, C22.45C24 [×2], C22.50C24 [×2], C42⋊8Q8
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=c2, 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)
(1 119 53 104)(2 118 54 103)(3 117 55 102)(4 120 56 101)(5 100 113 51)(6 99 114 50)(7 98 115 49)(8 97 116 52)(9 94 26 79)(10 93 27 78)(11 96 28 77)(12 95 25 80)(13 90 30 75)(14 89 31 74)(15 92 32 73)(16 91 29 76)(17 35 70 87)(18 34 71 86)(19 33 72 85)(20 36 69 88)(21 39 66 83)(22 38 67 82)(23 37 68 81)(24 40 65 84)(41 122 58 107)(42 121 59 106)(43 124 60 105)(44 123 57 108)(45 126 62 111)(46 125 63 110)(47 128 64 109)(48 127 61 112)
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), (1,119,53,104)(2,118,54,103)(3,117,55,102)(4,120,56,101)(5,100,113,51)(6,99,114,50)(7,98,115,49)(8,97,116,52)(9,94,26,79)(10,93,27,78)(11,96,28,77)(12,95,25,80)(13,90,30,75)(14,89,31,74)(15,92,32,73)(16,91,29,76)(17,35,70,87)(18,34,71,86)(19,33,72,85)(20,36,69,88)(21,39,66,83)(22,38,67,82)(23,37,68,81)(24,40,65,84)(41,122,58,107)(42,121,59,106)(43,124,60,105)(44,123,57,108)(45,126,62,111)(46,125,63,110)(47,128,64,109)(48,127,61,112)>;
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), (1,119,53,104)(2,118,54,103)(3,117,55,102)(4,120,56,101)(5,100,113,51)(6,99,114,50)(7,98,115,49)(8,97,116,52)(9,94,26,79)(10,93,27,78)(11,96,28,77)(12,95,25,80)(13,90,30,75)(14,89,31,74)(15,92,32,73)(16,91,29,76)(17,35,70,87)(18,34,71,86)(19,33,72,85)(20,36,69,88)(21,39,66,83)(22,38,67,82)(23,37,68,81)(24,40,65,84)(41,122,58,107)(42,121,59,106)(43,124,60,105)(44,123,57,108)(45,126,62,111)(46,125,63,110)(47,128,64,109)(48,127,61,112) );
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)], [(1,119,53,104),(2,118,54,103),(3,117,55,102),(4,120,56,101),(5,100,113,51),(6,99,114,50),(7,98,115,49),(8,97,116,52),(9,94,26,79),(10,93,27,78),(11,96,28,77),(12,95,25,80),(13,90,30,75),(14,89,31,74),(15,92,32,73),(16,91,29,76),(17,35,70,87),(18,34,71,86),(19,33,72,85),(20,36,69,88),(21,39,66,83),(22,38,67,82),(23,37,68,81),(24,40,65,84),(41,122,58,107),(42,121,59,106),(43,124,60,105),(44,123,57,108),(45,126,62,111),(46,125,63,110),(47,128,64,109),(48,127,61,112)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 0 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 3 |
G:=sub<GL(6,GF(5))| [0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,4,0,0,0,0,0,3],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,3,3] >;
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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42⋊8Q8 | C42⋊4C4 | C4×C4⋊C4 | C42⋊5C4 | C23.63C23 | C23.67C23 | C23.78C23 | C23.83C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 16 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_8Q_8
% in TeX
G:=Group("C4^2:8Q8"); // GroupNames label
G:=SmallGroup(128,1337);
// by ID
G=gap.SmallGroup(128,1337);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,184,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^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