direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42⋊5C4, C23.164C24, C24.642C23, C42⋊41(C2×C4), (C2×C42)⋊15C4, (C22×C42).10C2, C22.55(C23×C4), C23.356(C4○D4), (C23×C4).644C22, (C22×C4).442C23, C23.281(C22×C4), (C2×C42).1000C22, C22.66(C42⋊C2), C22.29(C42⋊2C2), C2.C42.463C22, C2.1(C2×C42⋊2C2), C22.57(C2×C4○D4), (C22×C4).452(C2×C4), (C2×C4).486(C22×C4), C2.10(C2×C42⋊C2), (C2×C2.C42).8C2, SmallGroup(128,1014)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 492 in 312 conjugacy classes, 180 normal (6 characteristic)
C1, C2 [×15], C4 [×20], C22, C22 [×34], C2×C4 [×12], C2×C4 [×76], C23, C23 [×14], C42 [×16], C22×C4 [×26], C22×C4 [×36], C24, C2.C42 [×24], C2×C42 [×12], C23×C4 [×7], C2×C2.C42 [×6], C42⋊5C4 [×8], C22×C42, C2×C42⋊5C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×12], C24, C42⋊C2 [×12], C42⋊2C2 [×16], C23×C4, C2×C4○D4 [×6], C42⋊5C4 [×8], C2×C42⋊C2 [×3], C2×C42⋊2C2 [×4], C2×C42⋊5C4
Generators and relations
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >
(1 15)(2 16)(3 13)(4 14)(5 111)(6 112)(7 109)(8 110)(9 20)(10 17)(11 18)(12 19)(21 98)(22 99)(23 100)(24 97)(25 80)(26 77)(27 78)(28 79)(29 76)(30 73)(31 74)(32 75)(33 60)(34 57)(35 58)(36 59)(37 64)(38 61)(39 62)(40 63)(41 91)(42 92)(43 89)(44 90)(45 95)(46 96)(47 93)(48 94)(49 53)(50 54)(51 55)(52 56)(65 86)(66 87)(67 88)(68 85)(69 82)(70 83)(71 84)(72 81)(101 122)(102 123)(103 124)(104 121)(105 118)(106 119)(107 120)(108 117)(113 126)(114 127)(115 128)(116 125)
(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 27 20 29)(2 28 17 30)(3 25 18 31)(4 26 19 32)(5 105 126 103)(6 106 127 104)(7 107 128 101)(8 108 125 102)(9 76 15 78)(10 73 16 79)(11 74 13 80)(12 75 14 77)(21 60 37 54)(22 57 38 55)(23 58 39 56)(24 59 40 53)(33 64 50 98)(34 61 51 99)(35 62 52 100)(36 63 49 97)(41 69 47 67)(42 70 48 68)(43 71 45 65)(44 72 46 66)(81 96 87 90)(82 93 88 91)(83 94 85 92)(84 95 86 89)(109 120 115 122)(110 117 116 123)(111 118 113 124)(112 119 114 121)
(1 65 36 119)(2 72 33 122)(3 67 34 117)(4 70 35 124)(5 73 94 21)(6 80 95 38)(7 75 96 23)(8 78 93 40)(9 84 53 104)(10 87 54 107)(11 82 55 102)(12 85 56 105)(13 88 57 108)(14 83 58 103)(15 86 59 106)(16 81 60 101)(17 66 50 120)(18 69 51 123)(19 68 52 118)(20 71 49 121)(22 127 74 89)(24 125 76 91)(25 45 61 112)(26 44 62 115)(27 47 63 110)(28 42 64 113)(29 41 97 116)(30 48 98 111)(31 43 99 114)(32 46 100 109)(37 126 79 92)(39 128 77 90)
G:=sub<Sym(128)| (1,15)(2,16)(3,13)(4,14)(5,111)(6,112)(7,109)(8,110)(9,20)(10,17)(11,18)(12,19)(21,98)(22,99)(23,100)(24,97)(25,80)(26,77)(27,78)(28,79)(29,76)(30,73)(31,74)(32,75)(33,60)(34,57)(35,58)(36,59)(37,64)(38,61)(39,62)(40,63)(41,91)(42,92)(43,89)(44,90)(45,95)(46,96)(47,93)(48,94)(49,53)(50,54)(51,55)(52,56)(65,86)(66,87)(67,88)(68,85)(69,82)(70,83)(71,84)(72,81)(101,122)(102,123)(103,124)(104,121)(105,118)(106,119)(107,120)(108,117)(113,126)(114,127)(115,128)(116,125), (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), (1,27,20,29)(2,28,17,30)(3,25,18,31)(4,26,19,32)(5,105,126,103)(6,106,127,104)(7,107,128,101)(8,108,125,102)(9,76,15,78)(10,73,16,79)(11,74,13,80)(12,75,14,77)(21,60,37,54)(22,57,38,55)(23,58,39,56)(24,59,40,53)(33,64,50,98)(34,61,51,99)(35,62,52,100)(36,63,49,97)(41,69,47,67)(42,70,48,68)(43,71,45,65)(44,72,46,66)(81,96,87,90)(82,93,88,91)(83,94,85,92)(84,95,86,89)(109,120,115,122)(110,117,116,123)(111,118,113,124)(112,119,114,121), (1,65,36,119)(2,72,33,122)(3,67,34,117)(4,70,35,124)(5,73,94,21)(6,80,95,38)(7,75,96,23)(8,78,93,40)(9,84,53,104)(10,87,54,107)(11,82,55,102)(12,85,56,105)(13,88,57,108)(14,83,58,103)(15,86,59,106)(16,81,60,101)(17,66,50,120)(18,69,51,123)(19,68,52,118)(20,71,49,121)(22,127,74,89)(24,125,76,91)(25,45,61,112)(26,44,62,115)(27,47,63,110)(28,42,64,113)(29,41,97,116)(30,48,98,111)(31,43,99,114)(32,46,100,109)(37,126,79,92)(39,128,77,90)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,111)(6,112)(7,109)(8,110)(9,20)(10,17)(11,18)(12,19)(21,98)(22,99)(23,100)(24,97)(25,80)(26,77)(27,78)(28,79)(29,76)(30,73)(31,74)(32,75)(33,60)(34,57)(35,58)(36,59)(37,64)(38,61)(39,62)(40,63)(41,91)(42,92)(43,89)(44,90)(45,95)(46,96)(47,93)(48,94)(49,53)(50,54)(51,55)(52,56)(65,86)(66,87)(67,88)(68,85)(69,82)(70,83)(71,84)(72,81)(101,122)(102,123)(103,124)(104,121)(105,118)(106,119)(107,120)(108,117)(113,126)(114,127)(115,128)(116,125), (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), (1,27,20,29)(2,28,17,30)(3,25,18,31)(4,26,19,32)(5,105,126,103)(6,106,127,104)(7,107,128,101)(8,108,125,102)(9,76,15,78)(10,73,16,79)(11,74,13,80)(12,75,14,77)(21,60,37,54)(22,57,38,55)(23,58,39,56)(24,59,40,53)(33,64,50,98)(34,61,51,99)(35,62,52,100)(36,63,49,97)(41,69,47,67)(42,70,48,68)(43,71,45,65)(44,72,46,66)(81,96,87,90)(82,93,88,91)(83,94,85,92)(84,95,86,89)(109,120,115,122)(110,117,116,123)(111,118,113,124)(112,119,114,121), (1,65,36,119)(2,72,33,122)(3,67,34,117)(4,70,35,124)(5,73,94,21)(6,80,95,38)(7,75,96,23)(8,78,93,40)(9,84,53,104)(10,87,54,107)(11,82,55,102)(12,85,56,105)(13,88,57,108)(14,83,58,103)(15,86,59,106)(16,81,60,101)(17,66,50,120)(18,69,51,123)(19,68,52,118)(20,71,49,121)(22,127,74,89)(24,125,76,91)(25,45,61,112)(26,44,62,115)(27,47,63,110)(28,42,64,113)(29,41,97,116)(30,48,98,111)(31,43,99,114)(32,46,100,109)(37,126,79,92)(39,128,77,90) );
G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,111),(6,112),(7,109),(8,110),(9,20),(10,17),(11,18),(12,19),(21,98),(22,99),(23,100),(24,97),(25,80),(26,77),(27,78),(28,79),(29,76),(30,73),(31,74),(32,75),(33,60),(34,57),(35,58),(36,59),(37,64),(38,61),(39,62),(40,63),(41,91),(42,92),(43,89),(44,90),(45,95),(46,96),(47,93),(48,94),(49,53),(50,54),(51,55),(52,56),(65,86),(66,87),(67,88),(68,85),(69,82),(70,83),(71,84),(72,81),(101,122),(102,123),(103,124),(104,121),(105,118),(106,119),(107,120),(108,117),(113,126),(114,127),(115,128),(116,125)], [(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)], [(1,27,20,29),(2,28,17,30),(3,25,18,31),(4,26,19,32),(5,105,126,103),(6,106,127,104),(7,107,128,101),(8,108,125,102),(9,76,15,78),(10,73,16,79),(11,74,13,80),(12,75,14,77),(21,60,37,54),(22,57,38,55),(23,58,39,56),(24,59,40,53),(33,64,50,98),(34,61,51,99),(35,62,52,100),(36,63,49,97),(41,69,47,67),(42,70,48,68),(43,71,45,65),(44,72,46,66),(81,96,87,90),(82,93,88,91),(83,94,85,92),(84,95,86,89),(109,120,115,122),(110,117,116,123),(111,118,113,124),(112,119,114,121)], [(1,65,36,119),(2,72,33,122),(3,67,34,117),(4,70,35,124),(5,73,94,21),(6,80,95,38),(7,75,96,23),(8,78,93,40),(9,84,53,104),(10,87,54,107),(11,82,55,102),(12,85,56,105),(13,88,57,108),(14,83,58,103),(15,86,59,106),(16,81,60,101),(17,66,50,120),(18,69,51,123),(19,68,52,118),(20,71,49,121),(22,127,74,89),(24,125,76,91),(25,45,61,112),(26,44,62,115),(27,47,63,110),(28,42,64,113),(29,41,97,116),(30,48,98,111),(31,43,99,114),(32,46,100,109),(37,126,79,92),(39,128,77,90)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X | 4Y | ··· | 4AN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | C4○D4 |
| kernel | C2×C42⋊5C4 | C2×C2.C42 | C42⋊5C4 | C22×C42 | C2×C42 | C23 |
| # reps | 1 | 6 | 8 | 1 | 16 | 24 |
In GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_5C_4
% in TeX
G:=Group("C2xC4^2:5C4"); // GroupNames label
G:=SmallGroup(128,1014);
// by ID
G=gap.SmallGroup(128,1014);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,100]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c^-1>;
// generators/relations