direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C8⋊C4, C23.43C42, C42.586C23, C23.37M4(2), C8⋊10(C22×C4), (C22×C8)⋊20C4, C4.39(C2×C42), (C2×C4).93C42, (C2×C42).40C4, C4.56(C23×C4), (C23×C8).24C2, (C23×C4).32C4, (C2×C4).620C24, (C2×C8).609C23, C42.299(C2×C4), C24.135(C2×C4), C22.34(C2×C42), C22.31(C23×C4), (C22×C42).12C2, C2.12(C22×C42), C2.1(C22×M4(2)), C23.286(C22×C4), (C23×C4).718C22, (C22×C8).581C22, C22.57(C2×M4(2)), (C2×C42).1021C22, (C22×C4).1646C23, C4○(C2×C8⋊C4), (C2×C8)⋊39(C2×C4), (C2×C4)○2(C8⋊C4), (C22×C4)○(C8⋊C4), (C22×C4).454(C2×C4), (C2×C4).495(C22×C4), (C2×C4)○(C2×C8⋊C4), (C22×C4)○(C2×C8⋊C4), SmallGroup(128,1602)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 380 in 340 conjugacy classes, 300 normal (8 characteristic)
C1, C2, C2 [×14], C4 [×8], C4 [×8], C22, C22 [×34], C8 [×16], C2×C4, C2×C4 [×35], C2×C4 [×24], C23 [×15], C42 [×16], C2×C8 [×56], C22×C4 [×26], C22×C4 [×8], C24, C8⋊C4 [×16], C2×C42 [×12], C22×C8 [×28], C23×C4, C23×C4 [×2], C2×C8⋊C4 [×12], C22×C42, C23×C8 [×2], C22×C8⋊C4
Quotients:
C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], M4(2) [×8], C22×C4 [×42], C24, C8⋊C4 [×16], C2×C42 [×12], C2×M4(2) [×12], C23×C4 [×3], C2×C8⋊C4 [×12], C22×C42, C22×M4(2) [×2], C22×C8⋊C4
Generators and relations
G = < a,b,c,d | a2=b2=c8=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
►Regular action on 128 points
Generators in S128
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 78)(10 79)(11 80)(12 73)(13 74)(14 75)(15 76)(16 77)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 105)(48 106)(57 92)(58 93)(59 94)(60 95)(61 96)(62 89)(63 90)(64 91)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(81 101)(82 102)(83 103)(84 104)(85 97)(86 98)(87 99)(88 100)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 41)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 89)(16 90)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 113)(24 114)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 97)(32 98)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 79)(58 80)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(105 124)(106 125)(107 126)(108 127)(109 128)(110 121)(111 122)(112 123)
(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 60 17 25)(2 57 18 30)(3 62 19 27)(4 59 20 32)(5 64 21 29)(6 61 22 26)(7 58 23 31)(8 63 24 28)(9 66 83 112)(10 71 84 109)(11 68 85 106)(12 65 86 111)(13 70 87 108)(14 67 88 105)(15 72 81 110)(16 69 82 107)(33 54 122 94)(34 51 123 91)(35 56 124 96)(36 53 125 93)(37 50 126 90)(38 55 127 95)(39 52 128 92)(40 49 121 89)(41 77 114 102)(42 74 115 99)(43 79 116 104)(44 76 117 101)(45 73 118 98)(46 78 119 103)(47 75 120 100)(48 80 113 97)
G:=sub<Sym(128)| (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,78)(10,79)(11,80)(12,73)(13,74)(14,75)(15,76)(16,77)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,105)(48,106)(57,92)(58,93)(59,94)(60,95)(61,96)(62,89)(63,90)(64,91)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(81,101)(82,102)(83,103)(84,104)(85,97)(86,98)(87,99)(88,100), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,41)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,113)(24,114)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,97)(32,98)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,79)(58,80)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(105,124)(106,125)(107,126)(108,127)(109,128)(110,121)(111,122)(112,123), (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), (1,60,17,25)(2,57,18,30)(3,62,19,27)(4,59,20,32)(5,64,21,29)(6,61,22,26)(7,58,23,31)(8,63,24,28)(9,66,83,112)(10,71,84,109)(11,68,85,106)(12,65,86,111)(13,70,87,108)(14,67,88,105)(15,72,81,110)(16,69,82,107)(33,54,122,94)(34,51,123,91)(35,56,124,96)(36,53,125,93)(37,50,126,90)(38,55,127,95)(39,52,128,92)(40,49,121,89)(41,77,114,102)(42,74,115,99)(43,79,116,104)(44,76,117,101)(45,73,118,98)(46,78,119,103)(47,75,120,100)(48,80,113,97)>;
G:=Group( (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,78)(10,79)(11,80)(12,73)(13,74)(14,75)(15,76)(16,77)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,105)(48,106)(57,92)(58,93)(59,94)(60,95)(61,96)(62,89)(63,90)(64,91)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(81,101)(82,102)(83,103)(84,104)(85,97)(86,98)(87,99)(88,100), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,41)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,113)(24,114)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,97)(32,98)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,79)(58,80)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(105,124)(106,125)(107,126)(108,127)(109,128)(110,121)(111,122)(112,123), (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), (1,60,17,25)(2,57,18,30)(3,62,19,27)(4,59,20,32)(5,64,21,29)(6,61,22,26)(7,58,23,31)(8,63,24,28)(9,66,83,112)(10,71,84,109)(11,68,85,106)(12,65,86,111)(13,70,87,108)(14,67,88,105)(15,72,81,110)(16,69,82,107)(33,54,122,94)(34,51,123,91)(35,56,124,96)(36,53,125,93)(37,50,126,90)(38,55,127,95)(39,52,128,92)(40,49,121,89)(41,77,114,102)(42,74,115,99)(43,79,116,104)(44,76,117,101)(45,73,118,98)(46,78,119,103)(47,75,120,100)(48,80,113,97) );
G=PermutationGroup([(1,127),(2,128),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,78),(10,79),(11,80),(12,73),(13,74),(14,75),(15,76),(16,77),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(41,107),(42,108),(43,109),(44,110),(45,111),(46,112),(47,105),(48,106),(57,92),(58,93),(59,94),(60,95),(61,96),(62,89),(63,90),(64,91),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(81,101),(82,102),(83,103),(84,104),(85,97),(86,98),(87,99),(88,100)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,41),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,89),(16,90),(17,115),(18,116),(19,117),(20,118),(21,119),(22,120),(23,113),(24,114),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,97),(32,98),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,79),(58,80),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(105,124),(106,125),(107,126),(108,127),(109,128),(110,121),(111,122),(112,123)], [(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)], [(1,60,17,25),(2,57,18,30),(3,62,19,27),(4,59,20,32),(5,64,21,29),(6,61,22,26),(7,58,23,31),(8,63,24,28),(9,66,83,112),(10,71,84,109),(11,68,85,106),(12,65,86,111),(13,70,87,108),(14,67,88,105),(15,72,81,110),(16,69,82,107),(33,54,122,94),(34,51,123,91),(35,56,124,96),(36,53,125,93),(37,50,126,90),(38,55,127,95),(39,52,128,92),(40,49,121,89),(41,77,114,102),(42,74,115,99),(43,79,116,104),(44,76,117,101),(45,73,118,98),(46,78,119,103),(47,75,120,100),(48,80,113,97)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 2 | 5 |
| 0 | 0 | 0 | 12 | 15 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,2,12,0,0,0,5,15],[4,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 4Q | ··· | 4AF | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) |
| kernel | C22×C8⋊C4 | C2×C8⋊C4 | C22×C42 | C23×C8 | C2×C42 | C22×C8 | C23×C4 | C23 |
| # reps | 1 | 12 | 1 | 2 | 12 | 32 | 4 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times C_8\rtimes C_4
% in TeX
G:=Group("C2^2xC8:C4"); // GroupNames label
G:=SmallGroup(128,1602);
// by ID
G=gap.SmallGroup(128,1602);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,925,232,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀