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), (C2×C42).555C22, (C22×C4).839C23, C22.301(C22×D4), C22.100(C22×Q8), (C22×Q8).133C22, C23.78C23.6C2, C23.65C23.52C2, C23.81C23.14C2, C2.C42.187C22, C23.67C23.39C2, C2.26(C23.37C23), C2.11(C23.41C23), C2.13(C2×C4⋊Q8), (C4×C4⋊C4).63C2, (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
Subgroups: 388 in 234 conjugacy classes, 120 normal (42 characteristic)
C1, C2 [×7], C4 [×4], C4 [×22], C22 [×7], C2×C4 [×22], C2×C4 [×34], Q8 [×8], C23, C42 [×4], C42 [×6], C4⋊C4 [×8], C4⋊C4 [×26], C22×C4 [×7], C22×C4 [×8], C2×Q8 [×10], C2.C42 [×2], C2.C42 [×10], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×6], C2×C4⋊C4 [×10], C42.C2 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C4×C4⋊C4 [×2], C42⋊8C4, C23.65C23 [×2], C23.65C23 [×2], C23.67C23 [×2], C23.78C23 [×2], C23.81C23 [×2], C2×C42.C2, C2×C4⋊Q8, C42⋊6Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ (1+4), 2- (1+4), C2×C4⋊Q8, C23.37C23, C23.41C23, D4⋊5D4, Q8⋊5D4, D4⋊3Q8, Q8⋊3Q8, C42⋊6Q8
Generators and relations
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 >
(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 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(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 45)(2 58 52 48)(3 57 49 47)(4 60 50 46)(5 42 62 56)(6 41 63 55)(7 44 64 54)(8 43 61 53)(9 31 23 17)(10 30 24 20)(11 29 21 19)(12 32 22 18)(13 71 27 39)(14 70 28 38)(15 69 25 37)(16 72 26 40)(33 105 98 95)(34 108 99 94)(35 107 100 93)(36 106 97 96)(65 117 125 75)(66 120 126 74)(67 119 127 73)(68 118 128 76)(77 85 115 123)(78 88 116 122)(79 87 113 121)(80 86 114 124)(81 89 103 111)(82 92 104 110)(83 91 101 109)(84 90 102 112)
(1 83 51 101)(2 116 52 78)(3 81 49 103)(4 114 50 80)(5 126 62 66)(6 33 63 98)(7 128 64 68)(8 35 61 100)(9 115 23 77)(10 84 24 102)(11 113 21 79)(12 82 22 104)(13 119 27 73)(14 96 28 106)(15 117 25 75)(16 94 26 108)(17 123 31 85)(18 92 32 110)(19 121 29 87)(20 90 30 112)(34 72 99 40)(36 70 97 38)(37 125 69 65)(39 127 71 67)(41 95 55 105)(42 120 56 74)(43 93 53 107)(44 118 54 76)(45 91 59 109)(46 124 60 86)(47 89 57 111)(48 122 58 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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(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), (1,59,51,45)(2,58,52,48)(3,57,49,47)(4,60,50,46)(5,42,62,56)(6,41,63,55)(7,44,64,54)(8,43,61,53)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(13,71,27,39)(14,70,28,38)(15,69,25,37)(16,72,26,40)(33,105,98,95)(34,108,99,94)(35,107,100,93)(36,106,97,96)(65,117,125,75)(66,120,126,74)(67,119,127,73)(68,118,128,76)(77,85,115,123)(78,88,116,122)(79,87,113,121)(80,86,114,124)(81,89,103,111)(82,92,104,110)(83,91,101,109)(84,90,102,112), (1,83,51,101)(2,116,52,78)(3,81,49,103)(4,114,50,80)(5,126,62,66)(6,33,63,98)(7,128,64,68)(8,35,61,100)(9,115,23,77)(10,84,24,102)(11,113,21,79)(12,82,22,104)(13,119,27,73)(14,96,28,106)(15,117,25,75)(16,94,26,108)(17,123,31,85)(18,92,32,110)(19,121,29,87)(20,90,30,112)(34,72,99,40)(36,70,97,38)(37,125,69,65)(39,127,71,67)(41,95,55,105)(42,120,56,74)(43,93,53,107)(44,118,54,76)(45,91,59,109)(46,124,60,86)(47,89,57,111)(48,122,58,88)>;
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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(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), (1,59,51,45)(2,58,52,48)(3,57,49,47)(4,60,50,46)(5,42,62,56)(6,41,63,55)(7,44,64,54)(8,43,61,53)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(13,71,27,39)(14,70,28,38)(15,69,25,37)(16,72,26,40)(33,105,98,95)(34,108,99,94)(35,107,100,93)(36,106,97,96)(65,117,125,75)(66,120,126,74)(67,119,127,73)(68,118,128,76)(77,85,115,123)(78,88,116,122)(79,87,113,121)(80,86,114,124)(81,89,103,111)(82,92,104,110)(83,91,101,109)(84,90,102,112), (1,83,51,101)(2,116,52,78)(3,81,49,103)(4,114,50,80)(5,126,62,66)(6,33,63,98)(7,128,64,68)(8,35,61,100)(9,115,23,77)(10,84,24,102)(11,113,21,79)(12,82,22,104)(13,119,27,73)(14,96,28,106)(15,117,25,75)(16,94,26,108)(17,123,31,85)(18,92,32,110)(19,121,29,87)(20,90,30,112)(34,72,99,40)(36,70,97,38)(37,125,69,65)(39,127,71,67)(41,95,55,105)(42,120,56,74)(43,93,53,107)(44,118,54,76)(45,91,59,109)(46,124,60,86)(47,89,57,111)(48,122,58,88) );
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)], [(1,13,9,41),(2,14,10,42),(3,15,11,43),(4,16,12,44),(5,48,38,20),(6,45,39,17),(7,46,40,18),(8,47,37,19),(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)], [(1,59,51,45),(2,58,52,48),(3,57,49,47),(4,60,50,46),(5,42,62,56),(6,41,63,55),(7,44,64,54),(8,43,61,53),(9,31,23,17),(10,30,24,20),(11,29,21,19),(12,32,22,18),(13,71,27,39),(14,70,28,38),(15,69,25,37),(16,72,26,40),(33,105,98,95),(34,108,99,94),(35,107,100,93),(36,106,97,96),(65,117,125,75),(66,120,126,74),(67,119,127,73),(68,118,128,76),(77,85,115,123),(78,88,116,122),(79,87,113,121),(80,86,114,124),(81,89,103,111),(82,92,104,110),(83,91,101,109),(84,90,102,112)], [(1,83,51,101),(2,116,52,78),(3,81,49,103),(4,114,50,80),(5,126,62,66),(6,33,63,98),(7,128,64,68),(8,35,61,100),(9,115,23,77),(10,84,24,102),(11,113,21,79),(12,82,22,104),(13,119,27,73),(14,96,28,106),(15,117,25,75),(16,94,26,108),(17,123,31,85),(18,92,32,110),(19,121,29,87),(20,90,30,112),(34,72,99,40),(36,70,97,38),(37,125,69,65),(39,127,71,67),(41,95,55,105),(42,120,56,74),(43,93,53,107),(44,118,54,76),(45,91,59,109),(46,124,60,86),(47,89,57,111),(48,122,58,88)])
Matrix representation ►G ⊆ 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] >;
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 |
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