p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊6Q8, C23.450C24, C22.2352+ 1+4, C22.1812- 1+4, C4⋊C4⋊18Q8, C4⋊C4.233D4, C4.26(C4⋊Q8), C2.72(D4⋊5D4), C42⋊8C4.32C2, C2.16(Q8⋊3Q8), C2.31(D4⋊3Q8), C2.39(Q8⋊5D4), (C22×C4).839C23, (C2×C42).555C22, C22.301(C22×D4), C22.100(C22×Q8), (C22×Q8).133C22, C23.78C23.6C2, C23.67C23.39C2, C2.C42.187C22, C23.65C23.52C2, C23.81C23.14C2, C2.26(C23.37C23), C2.11(C23.41C23), (C4×C4⋊C4).63C2, C2.13(C2×C4⋊Q8), (C2×C4).76(C2×D4), (C2×C4⋊Q8).32C2, (C2×C4).125(C2×Q8), (C2×C4).149(C4○D4), (C2×C4⋊C4).304C22, C22.327(C2×C4○D4), (C2×C42.C2).19C2, SmallGroup(128,1282)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊6Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 388 in 234 conjugacy classes, 120 normal (42 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C4⋊Q8, C22×Q8, C4×C4⋊C4, C42⋊8C4, C23.65C23, C23.65C23, C23.67C23, C23.78C23, C23.81C23, C2×C42.C2, C2×C4⋊Q8, C42⋊6Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4⋊Q8, C23.37C23, C23.41C23, D4⋊5D4, Q8⋊5D4, D4⋊3Q8, Q8⋊3Q8, C42⋊6Q8
►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 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 46 38 18)(6 47 39 19)(7 48 40 20)(8 45 37 17)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 123 127 91)(34 124 128 92)(35 121 125 89)(36 122 126 90)(65 111 100 87)(66 112 97 88)(67 109 98 85)(68 110 99 86)(73 101 105 77)(74 102 106 78)(75 103 107 79)(76 104 108 80)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)
(1 59 51 47)(2 58 52 46)(3 57 49 45)(4 60 50 48)(5 44 62 56)(6 43 63 55)(7 42 64 54)(8 41 61 53)(9 29 21 17)(10 32 22 20)(11 31 23 19)(12 30 24 18)(13 69 25 37)(14 72 26 40)(15 71 27 39)(16 70 28 38)(33 107 100 95)(34 106 97 94)(35 105 98 93)(36 108 99 96)(65 119 127 75)(66 118 128 74)(67 117 125 73)(68 120 126 76)(77 85 113 121)(78 88 114 124)(79 87 115 123)(80 86 116 122)(81 89 101 109)(82 92 102 112)(83 91 103 111)(84 90 104 110)
(1 83 51 103)(2 116 52 80)(3 81 49 101)(4 114 50 78)(5 126 62 68)(6 33 63 100)(7 128 64 66)(8 35 61 98)(9 113 21 77)(10 82 22 102)(11 115 23 79)(12 84 24 104)(13 117 25 73)(14 94 26 106)(15 119 27 75)(16 96 28 108)(17 121 29 85)(18 90 30 110)(19 123 31 87)(20 92 32 112)(34 72 97 40)(36 70 99 38)(37 125 69 67)(39 127 71 65)(41 93 53 105)(42 118 54 74)(43 95 55 107)(44 120 56 76)(45 89 57 109)(46 122 58 86)(47 91 59 111)(48 124 60 88)
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,46,38,18)(6,47,39,19)(7,48,40,20)(8,45,37,17)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,59,51,47)(2,58,52,46)(3,57,49,45)(4,60,50,48)(5,44,62,56)(6,43,63,55)(7,42,64,54)(8,41,61,53)(9,29,21,17)(10,32,22,20)(11,31,23,19)(12,30,24,18)(13,69,25,37)(14,72,26,40)(15,71,27,39)(16,70,28,38)(33,107,100,95)(34,106,97,94)(35,105,98,93)(36,108,99,96)(65,119,127,75)(66,118,128,74)(67,117,125,73)(68,120,126,76)(77,85,113,121)(78,88,114,124)(79,87,115,123)(80,86,116,122)(81,89,101,109)(82,92,102,112)(83,91,103,111)(84,90,104,110), (1,83,51,103)(2,116,52,80)(3,81,49,101)(4,114,50,78)(5,126,62,68)(6,33,63,100)(7,128,64,66)(8,35,61,98)(9,113,21,77)(10,82,22,102)(11,115,23,79)(12,84,24,104)(13,117,25,73)(14,94,26,106)(15,119,27,75)(16,96,28,108)(17,121,29,85)(18,90,30,110)(19,123,31,87)(20,92,32,112)(34,72,97,40)(36,70,99,38)(37,125,69,67)(39,127,71,65)(41,93,53,105)(42,118,54,74)(43,95,55,107)(44,120,56,76)(45,89,57,109)(46,122,58,86)(47,91,59,111)(48,124,60,88)>;
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,46,38,18)(6,47,39,19)(7,48,40,20)(8,45,37,17)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,59,51,47)(2,58,52,46)(3,57,49,45)(4,60,50,48)(5,44,62,56)(6,43,63,55)(7,42,64,54)(8,41,61,53)(9,29,21,17)(10,32,22,20)(11,31,23,19)(12,30,24,18)(13,69,25,37)(14,72,26,40)(15,71,27,39)(16,70,28,38)(33,107,100,95)(34,106,97,94)(35,105,98,93)(36,108,99,96)(65,119,127,75)(66,118,128,74)(67,117,125,73)(68,120,126,76)(77,85,113,121)(78,88,114,124)(79,87,115,123)(80,86,116,122)(81,89,101,109)(82,92,102,112)(83,91,103,111)(84,90,104,110), (1,83,51,103)(2,116,52,80)(3,81,49,101)(4,114,50,78)(5,126,62,68)(6,33,63,100)(7,128,64,66)(8,35,61,98)(9,113,21,77)(10,82,22,102)(11,115,23,79)(12,84,24,104)(13,117,25,73)(14,94,26,106)(15,119,27,75)(16,96,28,108)(17,121,29,85)(18,90,30,110)(19,123,31,87)(20,92,32,112)(34,72,97,40)(36,70,99,38)(37,125,69,67)(39,127,71,65)(41,93,53,105)(42,118,54,74)(43,95,55,107)(44,120,56,76)(45,89,57,109)(46,122,58,86)(47,91,59,111)(48,124,60,88) );
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)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,46,38,18),(6,47,39,19),(7,48,40,20),(8,45,37,17),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,69),(30,62,58,70),(31,63,59,71),(32,64,60,72),(33,123,127,91),(34,124,128,92),(35,121,125,89),(36,122,126,90),(65,111,100,87),(66,112,97,88),(67,109,98,85),(68,110,99,86),(73,101,105,77),(74,102,106,78),(75,103,107,79),(76,104,108,80),(81,93,113,117),(82,94,114,118),(83,95,115,119),(84,96,116,120)], [(1,59,51,47),(2,58,52,46),(3,57,49,45),(4,60,50,48),(5,44,62,56),(6,43,63,55),(7,42,64,54),(8,41,61,53),(9,29,21,17),(10,32,22,20),(11,31,23,19),(12,30,24,18),(13,69,25,37),(14,72,26,40),(15,71,27,39),(16,70,28,38),(33,107,100,95),(34,106,97,94),(35,105,98,93),(36,108,99,96),(65,119,127,75),(66,118,128,74),(67,117,125,73),(68,120,126,76),(77,85,113,121),(78,88,114,124),(79,87,115,123),(80,86,116,122),(81,89,101,109),(82,92,102,112),(83,91,103,111),(84,90,104,110)], [(1,83,51,103),(2,116,52,80),(3,81,49,101),(4,114,50,78),(5,126,62,68),(6,33,63,100),(7,128,64,66),(8,35,61,98),(9,113,21,77),(10,82,22,102),(11,115,23,79),(12,84,24,104),(13,117,25,73),(14,94,26,106),(15,119,27,75),(16,96,28,108),(17,121,29,85),(18,90,30,110),(19,123,31,87),(20,92,32,112),(34,72,97,40),(36,70,99,38),(37,125,69,67),(39,127,71,65),(41,93,53,105),(42,118,54,74),(43,95,55,107),(44,120,56,76),(45,89,57,109),(46,122,58,86),(47,91,59,111),(48,124,60,88)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC | 4AD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊6Q8 | C4×C4⋊C4 | C42⋊8C4 | C23.65C23 | C23.67C23 | C23.78C23 | C23.81C23 | C2×C42.C2 | C2×C4⋊Q8 | C42 | C4⋊C4 | C4⋊C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 4 | 8 | 1 | 1 |
Matrix representation of C42⋊6Q8 ►in GL6(𝔽5)
| 4 | 3 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
G:=sub<GL(6,GF(5))| [4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,4,0,0,0,0,0,0,3,2,0,0,0,0,0,2,0,0,0,0,0,0,3,3,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,4,0,0,0,0,2,1] >;
C42⋊6Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_6Q_8
% in TeX
G:=Group("C4^2:6Q8"); // GroupNames label
G:=SmallGroup(128,1282);
// by ID
G=gap.SmallGroup(128,1282);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations