p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4⋊1Q8, (C2×C8)⋊1Q8, C4⋊C4.99D4, (C2×C4).22D8, C2.6(C8⋊Q8), C4.21(C4⋊Q8), (C2×C4).17Q16, C2.4(C8⋊2Q8), C22.86(C2×D8), C4.6(C22⋊Q8), C2.7(C4.Q16), C2.7(D4⋊Q8), C23.919(C2×D4), (C22×C4).315D4, C22.48(C4⋊Q8), C2.23(C22⋊D8), C22.58(C2×Q16), C22.227C22≀C2, C22.4Q16.38C2, (C22×C8).114C22, (C2×C42).371C22, C2.23(C22⋊Q16), C22.142(C8⋊C22), (C22×C4).1453C23, C22.106(C22⋊Q8), C22.131(C8.C22), C22.7C42.24C2, C23.65C23.15C2, C2.4(C23.78C23), (C2×C4⋊Q8).21C2, (C2×C4).217(C2×Q8), (C2×C2.D8).12C2, (C2×C4).1048(C2×D4), (C2×C4).775(C4○D4), (C2×C4⋊C4).134C22, SmallGroup(128,789)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4⋊Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >
Subgroups: 288 in 141 conjugacy classes, 58 normal (28 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2.C42, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C23.65C23, C2×C2.D8, C2×C4⋊Q8, C4⋊C4⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D8, Q16, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4⋊Q8, C2×D8, C2×Q16, C8⋊C22, C8.C22, C23.78C23, C22⋊D8, C22⋊Q16, D4⋊Q8, C4.Q16, C8⋊2Q8, C8⋊Q8, C4⋊C4⋊Q8
►Regular action on 128 points
Generators in S128
(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 58 23 71)(2 57 24 70)(3 60 21 69)(4 59 22 72)(5 61 11 49)(6 64 12 52)(7 63 9 51)(8 62 10 50)(13 75 124 87)(14 74 121 86)(15 73 122 85)(16 76 123 88)(17 43 29 54)(18 42 30 53)(19 41 31 56)(20 44 32 55)(25 34 37 47)(26 33 38 46)(27 36 39 45)(28 35 40 48)(65 113 77 126)(66 116 78 125)(67 115 79 128)(68 114 80 127)(81 117 94 105)(82 120 95 108)(83 119 96 107)(84 118 93 106)(89 110 104 98)(90 109 101 97)(91 112 102 100)(92 111 103 99)
(1 53 8 33)(2 54 5 34)(3 55 6 35)(4 56 7 36)(9 45 22 41)(10 46 23 42)(11 47 24 43)(12 48 21 44)(13 103 115 83)(14 104 116 84)(15 101 113 81)(16 102 114 82)(17 49 37 70)(18 50 38 71)(19 51 39 72)(20 52 40 69)(25 57 29 61)(26 58 30 62)(27 59 31 63)(28 60 32 64)(65 117 85 97)(66 118 86 98)(67 119 87 99)(68 120 88 100)(73 109 77 105)(74 110 78 106)(75 111 79 107)(76 112 80 108)(89 125 93 121)(90 126 94 122)(91 127 95 123)(92 128 96 124)
(1 66 8 86)(2 65 5 85)(3 68 6 88)(4 67 7 87)(9 75 22 79)(10 74 23 78)(11 73 24 77)(12 76 21 80)(13 69 115 52)(14 72 116 51)(15 71 113 50)(16 70 114 49)(17 102 37 82)(18 101 38 81)(19 104 39 84)(20 103 40 83)(25 95 29 91)(26 94 30 90)(27 93 31 89)(28 96 32 92)(33 118 53 98)(34 117 54 97)(35 120 55 100)(36 119 56 99)(41 111 45 107)(42 110 46 106)(43 109 47 105)(44 112 48 108)(57 127 61 123)(58 126 62 122)(59 125 63 121)(60 128 64 124)
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), (1,58,23,71)(2,57,24,70)(3,60,21,69)(4,59,22,72)(5,61,11,49)(6,64,12,52)(7,63,9,51)(8,62,10,50)(13,75,124,87)(14,74,121,86)(15,73,122,85)(16,76,123,88)(17,43,29,54)(18,42,30,53)(19,41,31,56)(20,44,32,55)(25,34,37,47)(26,33,38,46)(27,36,39,45)(28,35,40,48)(65,113,77,126)(66,116,78,125)(67,115,79,128)(68,114,80,127)(81,117,94,105)(82,120,95,108)(83,119,96,107)(84,118,93,106)(89,110,104,98)(90,109,101,97)(91,112,102,100)(92,111,103,99), (1,53,8,33)(2,54,5,34)(3,55,6,35)(4,56,7,36)(9,45,22,41)(10,46,23,42)(11,47,24,43)(12,48,21,44)(13,103,115,83)(14,104,116,84)(15,101,113,81)(16,102,114,82)(17,49,37,70)(18,50,38,71)(19,51,39,72)(20,52,40,69)(25,57,29,61)(26,58,30,62)(27,59,31,63)(28,60,32,64)(65,117,85,97)(66,118,86,98)(67,119,87,99)(68,120,88,100)(73,109,77,105)(74,110,78,106)(75,111,79,107)(76,112,80,108)(89,125,93,121)(90,126,94,122)(91,127,95,123)(92,128,96,124), (1,66,8,86)(2,65,5,85)(3,68,6,88)(4,67,7,87)(9,75,22,79)(10,74,23,78)(11,73,24,77)(12,76,21,80)(13,69,115,52)(14,72,116,51)(15,71,113,50)(16,70,114,49)(17,102,37,82)(18,101,38,81)(19,104,39,84)(20,103,40,83)(25,95,29,91)(26,94,30,90)(27,93,31,89)(28,96,32,92)(33,118,53,98)(34,117,54,97)(35,120,55,100)(36,119,56,99)(41,111,45,107)(42,110,46,106)(43,109,47,105)(44,112,48,108)(57,127,61,123)(58,126,62,122)(59,125,63,121)(60,128,64,124)>;
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), (1,58,23,71)(2,57,24,70)(3,60,21,69)(4,59,22,72)(5,61,11,49)(6,64,12,52)(7,63,9,51)(8,62,10,50)(13,75,124,87)(14,74,121,86)(15,73,122,85)(16,76,123,88)(17,43,29,54)(18,42,30,53)(19,41,31,56)(20,44,32,55)(25,34,37,47)(26,33,38,46)(27,36,39,45)(28,35,40,48)(65,113,77,126)(66,116,78,125)(67,115,79,128)(68,114,80,127)(81,117,94,105)(82,120,95,108)(83,119,96,107)(84,118,93,106)(89,110,104,98)(90,109,101,97)(91,112,102,100)(92,111,103,99), (1,53,8,33)(2,54,5,34)(3,55,6,35)(4,56,7,36)(9,45,22,41)(10,46,23,42)(11,47,24,43)(12,48,21,44)(13,103,115,83)(14,104,116,84)(15,101,113,81)(16,102,114,82)(17,49,37,70)(18,50,38,71)(19,51,39,72)(20,52,40,69)(25,57,29,61)(26,58,30,62)(27,59,31,63)(28,60,32,64)(65,117,85,97)(66,118,86,98)(67,119,87,99)(68,120,88,100)(73,109,77,105)(74,110,78,106)(75,111,79,107)(76,112,80,108)(89,125,93,121)(90,126,94,122)(91,127,95,123)(92,128,96,124), (1,66,8,86)(2,65,5,85)(3,68,6,88)(4,67,7,87)(9,75,22,79)(10,74,23,78)(11,73,24,77)(12,76,21,80)(13,69,115,52)(14,72,116,51)(15,71,113,50)(16,70,114,49)(17,102,37,82)(18,101,38,81)(19,104,39,84)(20,103,40,83)(25,95,29,91)(26,94,30,90)(27,93,31,89)(28,96,32,92)(33,118,53,98)(34,117,54,97)(35,120,55,100)(36,119,56,99)(41,111,45,107)(42,110,46,106)(43,109,47,105)(44,112,48,108)(57,127,61,123)(58,126,62,122)(59,125,63,121)(60,128,64,124) );
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)], [(1,58,23,71),(2,57,24,70),(3,60,21,69),(4,59,22,72),(5,61,11,49),(6,64,12,52),(7,63,9,51),(8,62,10,50),(13,75,124,87),(14,74,121,86),(15,73,122,85),(16,76,123,88),(17,43,29,54),(18,42,30,53),(19,41,31,56),(20,44,32,55),(25,34,37,47),(26,33,38,46),(27,36,39,45),(28,35,40,48),(65,113,77,126),(66,116,78,125),(67,115,79,128),(68,114,80,127),(81,117,94,105),(82,120,95,108),(83,119,96,107),(84,118,93,106),(89,110,104,98),(90,109,101,97),(91,112,102,100),(92,111,103,99)], [(1,53,8,33),(2,54,5,34),(3,55,6,35),(4,56,7,36),(9,45,22,41),(10,46,23,42),(11,47,24,43),(12,48,21,44),(13,103,115,83),(14,104,116,84),(15,101,113,81),(16,102,114,82),(17,49,37,70),(18,50,38,71),(19,51,39,72),(20,52,40,69),(25,57,29,61),(26,58,30,62),(27,59,31,63),(28,60,32,64),(65,117,85,97),(66,118,86,98),(67,119,87,99),(68,120,88,100),(73,109,77,105),(74,110,78,106),(75,111,79,107),(76,112,80,108),(89,125,93,121),(90,126,94,122),(91,127,95,123),(92,128,96,124)], [(1,66,8,86),(2,65,5,85),(3,68,6,88),(4,67,7,87),(9,75,22,79),(10,74,23,78),(11,73,24,77),(12,76,21,80),(13,69,115,52),(14,72,116,51),(15,71,113,50),(16,70,114,49),(17,102,37,82),(18,101,38,81),(19,104,39,84),(20,103,40,83),(25,95,29,91),(26,94,30,90),(27,93,31,89),(28,96,32,92),(33,118,53,98),(34,117,54,97),(35,120,55,100),(36,119,56,99),(41,111,45,107),(42,110,46,106),(43,109,47,105),(44,112,48,108),(57,127,61,123),(58,126,62,122),(59,125,63,121),(60,128,64,124)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | D4 | D8 | Q16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C4⋊C4⋊Q8 | C22.7C42 | C22.4Q16 | C23.65C23 | C2×C2.D8 | C2×C4⋊Q8 | C4⋊C4 | C4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 4 | 2 | 1 | 1 |
Matrix representation of C4⋊C4⋊Q8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 11 | 8 | 0 | 0 | 0 | 0 |
| 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 13 |
| 0 | 0 | 0 | 0 | 13 | 11 |
| 10 | 15 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 7 | 0 | 0 | 0 | 0 |
| 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 4 |
| 0 | 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[11,2,0,0,0,0,8,6,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,6,13,0,0,0,0,13,11],[10,8,0,0,0,0,15,7,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,2,0,0,0,0,7,6,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,6,4,0,0,0,0,4,11] >;
C4⋊C4⋊Q8 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes Q_8
% in TeX
G:=Group("C4:C4:Q8"); // GroupNames label
G:=SmallGroup(128,789);
// by ID
G=gap.SmallGroup(128,789);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,58,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations