direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×Q8⋊3Q8, C22.65C25, C42.562C23, C23.277C24, C22.842- 1+4, Q8⋊6(C2×Q8), (C2×Q8)⋊17Q8, (C2×C4).63C24, C4.50(C22×Q8), C2.12(Q8×C23), C4⋊C4.474C23, C4⋊Q8.333C22, (C2×Q8).434C23, (C4×Q8).320C22, C22.52(C22×Q8), (C2×C42).933C22, C2.16(C2×2- 1+4), (C22×C4).1201C23, C42.C2.147C22, (C22×Q8).494C22, C4⋊C4○4(C2×Q8), Q8○3(C2×C4⋊C4), (C2×C4×Q8).55C2, (C2×C4⋊Q8).56C2, C4.175(C2×C4○D4), (C2×C4).324(C2×Q8), C2.37(C22×C4○D4), (C2×C4).908(C4○D4), (C2×C4⋊C4).702C22, C22.162(C2×C4○D4), (C2×C42.C2).37C2, (C2×Q8)○3(C2×C4⋊C4), SmallGroup(128,2208)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×Q8⋊3Q8
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b2c, ede-1=d-1 >
Subgroups: 604 in 524 conjugacy classes, 444 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C42.C2, C4⋊Q8, C22×Q8, C22×Q8, C2×C4×Q8, C2×C4×Q8, C2×C42.C2, C2×C4⋊Q8, Q8⋊3Q8, C2×Q8⋊3Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2- 1+4, C25, Q8⋊3Q8, Q8×C23, C22×C4○D4, C2×2- 1+4, C2×Q8⋊3Q8
►Regular action on 128 points
Generators in S128
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 113)(14 114)(15 115)(16 116)(17 38)(18 39)(19 40)(20 37)(25 32)(26 29)(27 30)(28 31)(33 54)(34 55)(35 56)(36 53)(41 46)(42 47)(43 48)(44 45)(49 72)(50 69)(51 70)(52 71)(57 62)(58 63)(59 64)(60 61)(65 88)(66 85)(67 86)(68 87)(73 80)(74 77)(75 78)(76 79)(81 102)(82 103)(83 104)(84 101)(89 96)(90 93)(91 94)(92 95)(97 118)(98 119)(99 120)(100 117)(105 110)(106 111)(107 112)(108 109)(121 126)(122 127)(123 128)(124 125)
(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 33 3 35)(2 36 4 34)(5 56 7 54)(6 55 8 53)(9 42 11 44)(10 41 12 43)(13 83 15 81)(14 82 16 84)(17 70 19 72)(18 69 20 71)(21 46 23 48)(22 45 24 47)(25 64 27 62)(26 63 28 61)(29 58 31 60)(30 57 32 59)(37 52 39 50)(38 51 40 49)(65 118 67 120)(66 117 68 119)(73 110 75 112)(74 109 76 111)(77 108 79 106)(78 107 80 105)(85 100 87 98)(86 99 88 97)(89 126 91 128)(90 125 92 127)(93 124 95 122)(94 123 96 121)(101 114 103 116)(102 113 104 115)
(1 17 24 31)(2 18 21 32)(3 19 22 29)(4 20 23 30)(5 40 11 26)(6 37 12 27)(7 38 9 28)(8 39 10 25)(13 99 122 109)(14 100 123 110)(15 97 124 111)(16 98 121 112)(33 70 47 60)(34 71 48 57)(35 72 45 58)(36 69 46 59)(41 64 53 50)(42 61 54 51)(43 62 55 52)(44 63 56 49)(65 90 79 104)(66 91 80 101)(67 92 77 102)(68 89 78 103)(73 84 85 94)(74 81 86 95)(75 82 87 96)(76 83 88 93)(105 114 117 128)(106 115 118 125)(107 116 119 126)(108 113 120 127)
(1 65 24 79)(2 66 21 80)(3 67 22 77)(4 68 23 78)(5 86 11 74)(6 87 12 75)(7 88 9 76)(8 85 10 73)(13 61 122 51)(14 62 123 52)(15 63 124 49)(16 64 121 50)(17 104 31 90)(18 101 32 91)(19 102 29 92)(20 103 30 89)(25 94 39 84)(26 95 40 81)(27 96 37 82)(28 93 38 83)(33 120 47 108)(34 117 48 105)(35 118 45 106)(36 119 46 107)(41 112 53 98)(42 109 54 99)(43 110 55 100)(44 111 56 97)(57 128 71 114)(58 125 72 115)(59 126 69 116)(60 127 70 113)
G:=sub<Sym(128)| (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,113)(14,114)(15,115)(16,116)(17,38)(18,39)(19,40)(20,37)(25,32)(26,29)(27,30)(28,31)(33,54)(34,55)(35,56)(36,53)(41,46)(42,47)(43,48)(44,45)(49,72)(50,69)(51,70)(52,71)(57,62)(58,63)(59,64)(60,61)(65,88)(66,85)(67,86)(68,87)(73,80)(74,77)(75,78)(76,79)(81,102)(82,103)(83,104)(84,101)(89,96)(90,93)(91,94)(92,95)(97,118)(98,119)(99,120)(100,117)(105,110)(106,111)(107,112)(108,109)(121,126)(122,127)(123,128)(124,125), (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), (1,33,3,35)(2,36,4,34)(5,56,7,54)(6,55,8,53)(9,42,11,44)(10,41,12,43)(13,83,15,81)(14,82,16,84)(17,70,19,72)(18,69,20,71)(21,46,23,48)(22,45,24,47)(25,64,27,62)(26,63,28,61)(29,58,31,60)(30,57,32,59)(37,52,39,50)(38,51,40,49)(65,118,67,120)(66,117,68,119)(73,110,75,112)(74,109,76,111)(77,108,79,106)(78,107,80,105)(85,100,87,98)(86,99,88,97)(89,126,91,128)(90,125,92,127)(93,124,95,122)(94,123,96,121)(101,114,103,116)(102,113,104,115), (1,17,24,31)(2,18,21,32)(3,19,22,29)(4,20,23,30)(5,40,11,26)(6,37,12,27)(7,38,9,28)(8,39,10,25)(13,99,122,109)(14,100,123,110)(15,97,124,111)(16,98,121,112)(33,70,47,60)(34,71,48,57)(35,72,45,58)(36,69,46,59)(41,64,53,50)(42,61,54,51)(43,62,55,52)(44,63,56,49)(65,90,79,104)(66,91,80,101)(67,92,77,102)(68,89,78,103)(73,84,85,94)(74,81,86,95)(75,82,87,96)(76,83,88,93)(105,114,117,128)(106,115,118,125)(107,116,119,126)(108,113,120,127), (1,65,24,79)(2,66,21,80)(3,67,22,77)(4,68,23,78)(5,86,11,74)(6,87,12,75)(7,88,9,76)(8,85,10,73)(13,61,122,51)(14,62,123,52)(15,63,124,49)(16,64,121,50)(17,104,31,90)(18,101,32,91)(19,102,29,92)(20,103,30,89)(25,94,39,84)(26,95,40,81)(27,96,37,82)(28,93,38,83)(33,120,47,108)(34,117,48,105)(35,118,45,106)(36,119,46,107)(41,112,53,98)(42,109,54,99)(43,110,55,100)(44,111,56,97)(57,128,71,114)(58,125,72,115)(59,126,69,116)(60,127,70,113)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,113)(14,114)(15,115)(16,116)(17,38)(18,39)(19,40)(20,37)(25,32)(26,29)(27,30)(28,31)(33,54)(34,55)(35,56)(36,53)(41,46)(42,47)(43,48)(44,45)(49,72)(50,69)(51,70)(52,71)(57,62)(58,63)(59,64)(60,61)(65,88)(66,85)(67,86)(68,87)(73,80)(74,77)(75,78)(76,79)(81,102)(82,103)(83,104)(84,101)(89,96)(90,93)(91,94)(92,95)(97,118)(98,119)(99,120)(100,117)(105,110)(106,111)(107,112)(108,109)(121,126)(122,127)(123,128)(124,125), (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), (1,33,3,35)(2,36,4,34)(5,56,7,54)(6,55,8,53)(9,42,11,44)(10,41,12,43)(13,83,15,81)(14,82,16,84)(17,70,19,72)(18,69,20,71)(21,46,23,48)(22,45,24,47)(25,64,27,62)(26,63,28,61)(29,58,31,60)(30,57,32,59)(37,52,39,50)(38,51,40,49)(65,118,67,120)(66,117,68,119)(73,110,75,112)(74,109,76,111)(77,108,79,106)(78,107,80,105)(85,100,87,98)(86,99,88,97)(89,126,91,128)(90,125,92,127)(93,124,95,122)(94,123,96,121)(101,114,103,116)(102,113,104,115), (1,17,24,31)(2,18,21,32)(3,19,22,29)(4,20,23,30)(5,40,11,26)(6,37,12,27)(7,38,9,28)(8,39,10,25)(13,99,122,109)(14,100,123,110)(15,97,124,111)(16,98,121,112)(33,70,47,60)(34,71,48,57)(35,72,45,58)(36,69,46,59)(41,64,53,50)(42,61,54,51)(43,62,55,52)(44,63,56,49)(65,90,79,104)(66,91,80,101)(67,92,77,102)(68,89,78,103)(73,84,85,94)(74,81,86,95)(75,82,87,96)(76,83,88,93)(105,114,117,128)(106,115,118,125)(107,116,119,126)(108,113,120,127), (1,65,24,79)(2,66,21,80)(3,67,22,77)(4,68,23,78)(5,86,11,74)(6,87,12,75)(7,88,9,76)(8,85,10,73)(13,61,122,51)(14,62,123,52)(15,63,124,49)(16,64,121,50)(17,104,31,90)(18,101,32,91)(19,102,29,92)(20,103,30,89)(25,94,39,84)(26,95,40,81)(27,96,37,82)(28,93,38,83)(33,120,47,108)(34,117,48,105)(35,118,45,106)(36,119,46,107)(41,112,53,98)(42,109,54,99)(43,110,55,100)(44,111,56,97)(57,128,71,114)(58,125,72,115)(59,126,69,116)(60,127,70,113) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6),(9,24),(10,21),(11,22),(12,23),(13,113),(14,114),(15,115),(16,116),(17,38),(18,39),(19,40),(20,37),(25,32),(26,29),(27,30),(28,31),(33,54),(34,55),(35,56),(36,53),(41,46),(42,47),(43,48),(44,45),(49,72),(50,69),(51,70),(52,71),(57,62),(58,63),(59,64),(60,61),(65,88),(66,85),(67,86),(68,87),(73,80),(74,77),(75,78),(76,79),(81,102),(82,103),(83,104),(84,101),(89,96),(90,93),(91,94),(92,95),(97,118),(98,119),(99,120),(100,117),(105,110),(106,111),(107,112),(108,109),(121,126),(122,127),(123,128),(124,125)], [(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)], [(1,33,3,35),(2,36,4,34),(5,56,7,54),(6,55,8,53),(9,42,11,44),(10,41,12,43),(13,83,15,81),(14,82,16,84),(17,70,19,72),(18,69,20,71),(21,46,23,48),(22,45,24,47),(25,64,27,62),(26,63,28,61),(29,58,31,60),(30,57,32,59),(37,52,39,50),(38,51,40,49),(65,118,67,120),(66,117,68,119),(73,110,75,112),(74,109,76,111),(77,108,79,106),(78,107,80,105),(85,100,87,98),(86,99,88,97),(89,126,91,128),(90,125,92,127),(93,124,95,122),(94,123,96,121),(101,114,103,116),(102,113,104,115)], [(1,17,24,31),(2,18,21,32),(3,19,22,29),(4,20,23,30),(5,40,11,26),(6,37,12,27),(7,38,9,28),(8,39,10,25),(13,99,122,109),(14,100,123,110),(15,97,124,111),(16,98,121,112),(33,70,47,60),(34,71,48,57),(35,72,45,58),(36,69,46,59),(41,64,53,50),(42,61,54,51),(43,62,55,52),(44,63,56,49),(65,90,79,104),(66,91,80,101),(67,92,77,102),(68,89,78,103),(73,84,85,94),(74,81,86,95),(75,82,87,96),(76,83,88,93),(105,114,117,128),(106,115,118,125),(107,116,119,126),(108,113,120,127)], [(1,65,24,79),(2,66,21,80),(3,67,22,77),(4,68,23,78),(5,86,11,74),(6,87,12,75),(7,88,9,76),(8,85,10,73),(13,61,122,51),(14,62,123,52),(15,63,124,49),(16,64,121,50),(17,104,31,90),(18,101,32,91),(19,102,29,92),(20,103,30,89),(25,94,39,84),(26,95,40,81),(27,96,37,82),(28,93,38,83),(33,120,47,108),(34,117,48,105),(35,118,45,106),(36,119,46,107),(41,112,53,98),(42,109,54,99),(43,110,55,100),(44,111,56,97),(57,128,71,114),(58,125,72,115),(59,126,69,116),(60,127,70,113)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 4Y | ··· | 4AP |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2- 1+4 |
| kernel | C2×Q8⋊3Q8 | C2×C4×Q8 | C2×C42.C2 | C2×C4⋊Q8 | Q8⋊3Q8 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 6 | 6 | 3 | 16 | 8 | 8 | 2 |
Matrix representation of C2×Q8⋊3Q8 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,3,2,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,4,0,0,0,2,4],[1,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,0,1,4,0,0,0,0,4] >;
C2×Q8⋊3Q8 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_3Q_8
% in TeX
G:=Group("C2xQ8:3Q8"); // GroupNames label
G:=SmallGroup(128,2208);
// by ID
G=gap.SmallGroup(128,2208);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,1430,352,570,136]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^4=1,c^2=b^2,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations