p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊18Q8, C43.16C2, C42.347D4, C23.762C24, C4.11(C4⋊Q8), C42⋊8C4.54C2, C4.17(C4.4D4), (C22×C4).267C23, C22.472(C22×D4), C22.183(C22×Q8), (C2×C42).1096C22, (C22×Q8).252C22, C2.C42.457C22, C23.67C23.65C2, C2.49(C23.37C23), C2.23(C2×C4⋊Q8), (C2×C4⋊Q8).40C2, (C2×C4).836(C2×D4), (C2×C4).174(C2×Q8), C2.36(C2×C4.4D4), (C2×C4).676(C4○D4), (C2×C4⋊C4).565C22, C22.603(C2×C4○D4), SmallGroup(128,1594)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 420 in 252 conjugacy classes, 132 normal (8 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×16], C22, C22 [×6], C2×C4 [×26], C2×C4 [×32], Q8 [×16], C23, C42 [×12], C42 [×8], C4⋊C4 [×16], C22×C4, C22×C4 [×14], C2×Q8 [×24], C2.C42 [×16], C2×C42, C2×C42 [×6], C2×C4⋊C4 [×8], C4⋊Q8 [×8], C22×Q8 [×4], C43, C42⋊8C4 [×4], C23.67C23 [×8], C2×C4⋊Q8 [×2], C42⋊18Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×8], C24, C4.4D4 [×8], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×4], C2×C4.4D4 [×2], C2×C4⋊Q8, C23.37C23 [×4], C42⋊18Q8
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >
►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 74 24 29)(2 75 21 30)(3 76 22 31)(4 73 23 32)(5 67 104 86)(6 68 101 87)(7 65 102 88)(8 66 103 85)(9 78 56 39)(10 79 53 40)(11 80 54 37)(12 77 55 38)(13 63 60 28)(14 64 57 25)(15 61 58 26)(16 62 59 27)(17 50 36 98)(18 51 33 99)(19 52 34 100)(20 49 35 97)(41 108 95 83)(42 105 96 84)(43 106 93 81)(44 107 94 82)(45 115 112 120)(46 116 109 117)(47 113 110 118)(48 114 111 119)(69 122 127 90)(70 123 128 91)(71 124 125 92)(72 121 126 89)
(1 100 60 38)(2 97 57 39)(3 98 58 40)(4 99 59 37)(5 108 111 122)(6 105 112 123)(7 106 109 124)(8 107 110 121)(9 75 20 25)(10 76 17 26)(11 73 18 27)(12 74 19 28)(13 77 24 52)(14 78 21 49)(15 79 22 50)(16 80 23 51)(29 34 63 55)(30 35 64 56)(31 36 61 53)(32 33 62 54)(41 114 69 86)(42 115 70 87)(43 116 71 88)(44 113 72 85)(45 91 101 84)(46 92 102 81)(47 89 103 82)(48 90 104 83)(65 93 117 125)(66 94 118 126)(67 95 119 127)(68 96 120 128)
(1 122 60 108)(2 121 57 107)(3 124 58 106)(4 123 59 105)(5 100 111 38)(6 99 112 37)(7 98 109 40)(8 97 110 39)(9 87 20 115)(10 86 17 114)(11 85 18 113)(12 88 19 116)(13 83 24 90)(14 82 21 89)(15 81 22 92)(16 84 23 91)(25 42 75 70)(26 41 76 69)(27 44 73 72)(28 43 74 71)(29 125 63 93)(30 128 64 96)(31 127 61 95)(32 126 62 94)(33 118 54 66)(34 117 55 65)(35 120 56 68)(36 119 53 67)(45 80 101 51)(46 79 102 50)(47 78 103 49)(48 77 104 52)
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), (1,74,24,29)(2,75,21,30)(3,76,22,31)(4,73,23,32)(5,67,104,86)(6,68,101,87)(7,65,102,88)(8,66,103,85)(9,78,56,39)(10,79,53,40)(11,80,54,37)(12,77,55,38)(13,63,60,28)(14,64,57,25)(15,61,58,26)(16,62,59,27)(17,50,36,98)(18,51,33,99)(19,52,34,100)(20,49,35,97)(41,108,95,83)(42,105,96,84)(43,106,93,81)(44,107,94,82)(45,115,112,120)(46,116,109,117)(47,113,110,118)(48,114,111,119)(69,122,127,90)(70,123,128,91)(71,124,125,92)(72,121,126,89), (1,100,60,38)(2,97,57,39)(3,98,58,40)(4,99,59,37)(5,108,111,122)(6,105,112,123)(7,106,109,124)(8,107,110,121)(9,75,20,25)(10,76,17,26)(11,73,18,27)(12,74,19,28)(13,77,24,52)(14,78,21,49)(15,79,22,50)(16,80,23,51)(29,34,63,55)(30,35,64,56)(31,36,61,53)(32,33,62,54)(41,114,69,86)(42,115,70,87)(43,116,71,88)(44,113,72,85)(45,91,101,84)(46,92,102,81)(47,89,103,82)(48,90,104,83)(65,93,117,125)(66,94,118,126)(67,95,119,127)(68,96,120,128), (1,122,60,108)(2,121,57,107)(3,124,58,106)(4,123,59,105)(5,100,111,38)(6,99,112,37)(7,98,109,40)(8,97,110,39)(9,87,20,115)(10,86,17,114)(11,85,18,113)(12,88,19,116)(13,83,24,90)(14,82,21,89)(15,81,22,92)(16,84,23,91)(25,42,75,70)(26,41,76,69)(27,44,73,72)(28,43,74,71)(29,125,63,93)(30,128,64,96)(31,127,61,95)(32,126,62,94)(33,118,54,66)(34,117,55,65)(35,120,56,68)(36,119,53,67)(45,80,101,51)(46,79,102,50)(47,78,103,49)(48,77,104,52)>;
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), (1,74,24,29)(2,75,21,30)(3,76,22,31)(4,73,23,32)(5,67,104,86)(6,68,101,87)(7,65,102,88)(8,66,103,85)(9,78,56,39)(10,79,53,40)(11,80,54,37)(12,77,55,38)(13,63,60,28)(14,64,57,25)(15,61,58,26)(16,62,59,27)(17,50,36,98)(18,51,33,99)(19,52,34,100)(20,49,35,97)(41,108,95,83)(42,105,96,84)(43,106,93,81)(44,107,94,82)(45,115,112,120)(46,116,109,117)(47,113,110,118)(48,114,111,119)(69,122,127,90)(70,123,128,91)(71,124,125,92)(72,121,126,89), (1,100,60,38)(2,97,57,39)(3,98,58,40)(4,99,59,37)(5,108,111,122)(6,105,112,123)(7,106,109,124)(8,107,110,121)(9,75,20,25)(10,76,17,26)(11,73,18,27)(12,74,19,28)(13,77,24,52)(14,78,21,49)(15,79,22,50)(16,80,23,51)(29,34,63,55)(30,35,64,56)(31,36,61,53)(32,33,62,54)(41,114,69,86)(42,115,70,87)(43,116,71,88)(44,113,72,85)(45,91,101,84)(46,92,102,81)(47,89,103,82)(48,90,104,83)(65,93,117,125)(66,94,118,126)(67,95,119,127)(68,96,120,128), (1,122,60,108)(2,121,57,107)(3,124,58,106)(4,123,59,105)(5,100,111,38)(6,99,112,37)(7,98,109,40)(8,97,110,39)(9,87,20,115)(10,86,17,114)(11,85,18,113)(12,88,19,116)(13,83,24,90)(14,82,21,89)(15,81,22,92)(16,84,23,91)(25,42,75,70)(26,41,76,69)(27,44,73,72)(28,43,74,71)(29,125,63,93)(30,128,64,96)(31,127,61,95)(32,126,62,94)(33,118,54,66)(34,117,55,65)(35,120,56,68)(36,119,53,67)(45,80,101,51)(46,79,102,50)(47,78,103,49)(48,77,104,52) );
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)], [(1,74,24,29),(2,75,21,30),(3,76,22,31),(4,73,23,32),(5,67,104,86),(6,68,101,87),(7,65,102,88),(8,66,103,85),(9,78,56,39),(10,79,53,40),(11,80,54,37),(12,77,55,38),(13,63,60,28),(14,64,57,25),(15,61,58,26),(16,62,59,27),(17,50,36,98),(18,51,33,99),(19,52,34,100),(20,49,35,97),(41,108,95,83),(42,105,96,84),(43,106,93,81),(44,107,94,82),(45,115,112,120),(46,116,109,117),(47,113,110,118),(48,114,111,119),(69,122,127,90),(70,123,128,91),(71,124,125,92),(72,121,126,89)], [(1,100,60,38),(2,97,57,39),(3,98,58,40),(4,99,59,37),(5,108,111,122),(6,105,112,123),(7,106,109,124),(8,107,110,121),(9,75,20,25),(10,76,17,26),(11,73,18,27),(12,74,19,28),(13,77,24,52),(14,78,21,49),(15,79,22,50),(16,80,23,51),(29,34,63,55),(30,35,64,56),(31,36,61,53),(32,33,62,54),(41,114,69,86),(42,115,70,87),(43,116,71,88),(44,113,72,85),(45,91,101,84),(46,92,102,81),(47,89,103,82),(48,90,104,83),(65,93,117,125),(66,94,118,126),(67,95,119,127),(68,96,120,128)], [(1,122,60,108),(2,121,57,107),(3,124,58,106),(4,123,59,105),(5,100,111,38),(6,99,112,37),(7,98,109,40),(8,97,110,39),(9,87,20,115),(10,86,17,114),(11,85,18,113),(12,88,19,116),(13,83,24,90),(14,82,21,89),(15,81,22,92),(16,84,23,91),(25,42,75,70),(26,41,76,69),(27,44,73,72),(28,43,74,71),(29,125,63,93),(30,128,64,96),(31,127,61,95),(32,126,62,94),(33,118,54,66),(34,117,55,65),(35,120,56,68),(36,119,53,67),(45,80,101,51),(46,79,102,50),(47,78,103,49),(48,77,104,52)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4AB | 4AC | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 |
| kernel | C42⋊18Q8 | C43 | C42⋊8C4 | C23.67C23 | C2×C4⋊Q8 | C42 | C42 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 4 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{18}Q_8 % in TeX
G:=Group("C4^2:18Q8"); // GroupNames label
G:=SmallGroup(128,1594);
// by ID
G=gap.SmallGroup(128,1594);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,184,2019,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀