p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.7M5(2), (C2×C4)⋊2C16, (C2×C16)⋊9C4, (C2×C8).9C8, C2.2(C4×C16), C2.1(C4⋊C16), C8.40(C4⋊C4), C4.23(C4⋊C8), (C2×C8).64Q8, (C2×C8).398D4, (C22×C4).7C8, (C2×C4).83C42, C22.14(C4×C8), (C2×C42).27C4, C23.42(C2×C8), C22.6(C2×C16), (C22×C8).23C4, (C22×C16).1C2, C2.3(C16⋊5C4), C4.15(C8⋊C4), C22.18(C4⋊C8), C8.54(C22⋊C4), C2.1(C22⋊C16), C4.30(C22⋊C8), (C2×C4).86M4(2), C22.37(C22⋊C8), (C22×C8).588C22, C4.24(C2.C42), C2.2(C22.7C42), (C2×C4×C8).5C2, (C2×C4).94(C2×C8), (C2×C8).258(C2×C4), (C2×C4).158(C4⋊C4), (C22×C4).501(C2×C4), (C2×C4).384(C22⋊C4), SmallGroup(128,106)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.7M5(2)
G = < a,b,c,d | a2=b2=c16=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=ac9 >
Subgroups: 104 in 80 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, C2×C16, C2×C42, C22×C8, C2×C4×C8, C22×C16, C22.7M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C16, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C16, M5(2), C22.7C42, C4×C16, C16⋊5C4, C22⋊C16, C4⋊C16, C22.7M5(2)
►Regular action on 128 points
Generators in S128
(1 111)(2 112)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(49 120)(50 121)(51 122)(52 123)(53 124)(54 125)(55 126)(56 127)(57 128)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 93)(66 94)(67 95)(68 96)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 127)(18 128)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 62)(34 63)(35 64)(36 49)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 61)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 97)(91 98)(92 99)(93 100)(94 101)(95 102)(96 103)
(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 63 76 34)(2 127 77 17)(3 49 78 36)(4 113 79 19)(5 51 80 38)(6 115 65 21)(7 53 66 40)(8 117 67 23)(9 55 68 42)(10 119 69 25)(11 57 70 44)(12 121 71 27)(13 59 72 46)(14 123 73 29)(15 61 74 48)(16 125 75 31)(18 105 128 82)(20 107 114 84)(22 109 116 86)(24 111 118 88)(26 97 120 90)(28 99 122 92)(30 101 124 94)(32 103 126 96)(33 102 62 95)(35 104 64 81)(37 106 50 83)(39 108 52 85)(41 110 54 87)(43 112 56 89)(45 98 58 91)(47 100 60 93)
G:=sub<Sym(128)| (1,111)(2,112)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(49,120)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,93)(66,94)(67,95)(68,96)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92), (1,76)(2,77)(3,78)(4,79)(5,80)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,127)(18,128)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,62)(34,63)(35,64)(36,49)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,61)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,97)(91,98)(92,99)(93,100)(94,101)(95,102)(96,103), (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), (1,63,76,34)(2,127,77,17)(3,49,78,36)(4,113,79,19)(5,51,80,38)(6,115,65,21)(7,53,66,40)(8,117,67,23)(9,55,68,42)(10,119,69,25)(11,57,70,44)(12,121,71,27)(13,59,72,46)(14,123,73,29)(15,61,74,48)(16,125,75,31)(18,105,128,82)(20,107,114,84)(22,109,116,86)(24,111,118,88)(26,97,120,90)(28,99,122,92)(30,101,124,94)(32,103,126,96)(33,102,62,95)(35,104,64,81)(37,106,50,83)(39,108,52,85)(41,110,54,87)(43,112,56,89)(45,98,58,91)(47,100,60,93)>;
G:=Group( (1,111)(2,112)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(49,120)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,93)(66,94)(67,95)(68,96)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92), (1,76)(2,77)(3,78)(4,79)(5,80)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,127)(18,128)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,62)(34,63)(35,64)(36,49)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,61)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,97)(91,98)(92,99)(93,100)(94,101)(95,102)(96,103), (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), (1,63,76,34)(2,127,77,17)(3,49,78,36)(4,113,79,19)(5,51,80,38)(6,115,65,21)(7,53,66,40)(8,117,67,23)(9,55,68,42)(10,119,69,25)(11,57,70,44)(12,121,71,27)(13,59,72,46)(14,123,73,29)(15,61,74,48)(16,125,75,31)(18,105,128,82)(20,107,114,84)(22,109,116,86)(24,111,118,88)(26,97,120,90)(28,99,122,92)(30,101,124,94)(32,103,126,96)(33,102,62,95)(35,104,64,81)(37,106,50,83)(39,108,52,85)(41,110,54,87)(43,112,56,89)(45,98,58,91)(47,100,60,93) );
G=PermutationGroup([[(1,111),(2,112),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,103),(10,104),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(49,120),(50,121),(51,122),(52,123),(53,124),(54,125),(55,126),(56,127),(57,128),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,119),(65,93),(66,94),(67,95),(68,96),(69,81),(70,82),(71,83),(72,84),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,73),(15,74),(16,75),(17,127),(18,128),(19,113),(20,114),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,62),(34,63),(35,64),(36,49),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,61),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,97),(91,98),(92,99),(93,100),(94,101),(95,102),(96,103)], [(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)], [(1,63,76,34),(2,127,77,17),(3,49,78,36),(4,113,79,19),(5,51,80,38),(6,115,65,21),(7,53,66,40),(8,117,67,23),(9,55,68,42),(10,119,69,25),(11,57,70,44),(12,121,71,27),(13,59,72,46),(14,123,73,29),(15,61,74,48),(16,125,75,31),(18,105,128,82),(20,107,114,84),(22,109,116,86),(24,111,118,88),(26,97,120,90),(28,99,122,92),(30,101,124,94),(32,103,126,96),(33,102,62,95),(35,104,64,81),(37,106,50,83),(39,108,52,85),(41,110,54,87),(43,112,56,89),(45,98,58,91),(47,100,60,93)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P | 8Q | ··· | 8X | 16A | ··· | 16AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C16 | D4 | Q8 | M4(2) | M5(2) |
| kernel | C22.7M5(2) | C2×C4×C8 | C22×C16 | C2×C16 | C2×C42 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C8 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 32 | 3 | 1 | 4 | 8 |
Matrix representation of C22.7M5(2) ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 3 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 8 | 16 |
| 0 | 0 | 14 | 9 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 2 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[3,0,0,0,0,13,0,0,0,0,8,14,0,0,16,9],[4,0,0,0,0,13,0,0,0,0,1,2,0,0,16,16] >;
C22.7M5(2) in GAP, Magma, Sage, TeX
C_2^2._7M_5(2)
% in TeX
G:=Group("C2^2.7M5(2)"); // GroupNames label
G:=SmallGroup(128,106);
// by ID
G=gap.SmallGroup(128,106);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,136,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^9>;
// generators/relations