p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊11Q8, C23.566C24, C22.2552- (1+4), C22.3402+ (1+4), C42⋊8C4.41C2, C42⋊5C4.13C2, (C2×C42).630C22, (C22×C4).171C23, C22.140(C22×Q8), (C22×Q8).171C22, C2.55(C22.32C24), C23.81C23.30C2, C23.78C23.17C2, C23.83C23.28C2, C23.67C23.51C2, C23.63C23.39C2, C2.C42.280C22, C23.65C23.70C2, C2.66(C22.36C24), C2.55(C22.33C24), C2.27(C23.41C23), C2.43(C23.37C23), C2.37(C22.35C24), (C4×C4⋊C4).79C2, (C2×C4).136(C2×Q8), (C2×C4).186(C4○D4), (C2×C4⋊C4).387C22, C22.433(C2×C4○D4), SmallGroup(128,1398)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 184 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C4 [×20], C22 [×7], C2×C4 [×10], C2×C4 [×40], Q8 [×4], C23, C42 [×4], C42 [×2], C4⋊C4 [×18], C22×C4 [×15], C2×Q8 [×4], C2.C42 [×18], C2×C42 [×3], C2×C4⋊C4 [×13], C22×Q8, C4×C4⋊C4, C42⋊8C4, C42⋊5C4, C23.63C23 [×2], C23.65C23, C23.67C23, C23.78C23 [×2], C23.81C23 [×4], C23.83C23 [×2], C42⋊11Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C23.37C23, C22.32C24, C22.33C24, C22.35C24, C22.36C24, C23.41C23 [×2], C42⋊11Q8
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b, bd=db, 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 13 107 44)(2 14 108 41)(3 15 105 42)(4 16 106 43)(5 66 93 40)(6 67 94 37)(7 68 95 38)(8 65 96 39)(9 103 48 76)(10 104 45 73)(11 101 46 74)(12 102 47 75)(17 109 56 82)(18 110 53 83)(19 111 54 84)(20 112 55 81)(21 113 52 78)(22 114 49 79)(23 115 50 80)(24 116 51 77)(25 119 64 92)(26 120 61 89)(27 117 62 90)(28 118 63 91)(29 123 60 88)(30 124 57 85)(31 121 58 86)(32 122 59 87)(33 125 71 98)(34 126 72 99)(35 127 69 100)(36 128 70 97)
(1 109 101 80)(2 83 102 116)(3 111 103 78)(4 81 104 114)(5 118 99 85)(6 92 100 121)(7 120 97 87)(8 90 98 123)(9 50 42 17)(10 24 43 53)(11 52 44 19)(12 22 41 55)(13 54 46 21)(14 20 47 49)(15 56 48 23)(16 18 45 51)(25 33 58 65)(26 72 59 40)(27 35 60 67)(28 70 57 38)(29 37 62 69)(30 68 63 36)(31 39 64 71)(32 66 61 34)(73 79 106 112)(74 115 107 82)(75 77 108 110)(76 113 105 84)(86 94 119 127)(88 96 117 125)(89 128 122 95)(91 126 124 93)
(1 117 101 88)(2 120 102 87)(3 119 103 86)(4 118 104 85)(5 114 99 81)(6 113 100 84)(7 116 97 83)(8 115 98 82)(9 58 42 25)(10 57 43 28)(11 60 44 27)(12 59 41 26)(13 62 46 29)(14 61 47 32)(15 64 48 31)(16 63 45 30)(17 65 50 33)(18 68 51 36)(19 67 52 35)(20 66 49 34)(21 69 54 37)(22 72 55 40)(23 71 56 39)(24 70 53 38)(73 124 106 91)(74 123 107 90)(75 122 108 89)(76 121 105 92)(77 128 110 95)(78 127 111 94)(79 126 112 93)(80 125 109 96)
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), (1,13,107,44)(2,14,108,41)(3,15,105,42)(4,16,106,43)(5,66,93,40)(6,67,94,37)(7,68,95,38)(8,65,96,39)(9,103,48,76)(10,104,45,73)(11,101,46,74)(12,102,47,75)(17,109,56,82)(18,110,53,83)(19,111,54,84)(20,112,55,81)(21,113,52,78)(22,114,49,79)(23,115,50,80)(24,116,51,77)(25,119,64,92)(26,120,61,89)(27,117,62,90)(28,118,63,91)(29,123,60,88)(30,124,57,85)(31,121,58,86)(32,122,59,87)(33,125,71,98)(34,126,72,99)(35,127,69,100)(36,128,70,97), (1,109,101,80)(2,83,102,116)(3,111,103,78)(4,81,104,114)(5,118,99,85)(6,92,100,121)(7,120,97,87)(8,90,98,123)(9,50,42,17)(10,24,43,53)(11,52,44,19)(12,22,41,55)(13,54,46,21)(14,20,47,49)(15,56,48,23)(16,18,45,51)(25,33,58,65)(26,72,59,40)(27,35,60,67)(28,70,57,38)(29,37,62,69)(30,68,63,36)(31,39,64,71)(32,66,61,34)(73,79,106,112)(74,115,107,82)(75,77,108,110)(76,113,105,84)(86,94,119,127)(88,96,117,125)(89,128,122,95)(91,126,124,93), (1,117,101,88)(2,120,102,87)(3,119,103,86)(4,118,104,85)(5,114,99,81)(6,113,100,84)(7,116,97,83)(8,115,98,82)(9,58,42,25)(10,57,43,28)(11,60,44,27)(12,59,41,26)(13,62,46,29)(14,61,47,32)(15,64,48,31)(16,63,45,30)(17,65,50,33)(18,68,51,36)(19,67,52,35)(20,66,49,34)(21,69,54,37)(22,72,55,40)(23,71,56,39)(24,70,53,38)(73,124,106,91)(74,123,107,90)(75,122,108,89)(76,121,105,92)(77,128,110,95)(78,127,111,94)(79,126,112,93)(80,125,109,96)>;
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), (1,13,107,44)(2,14,108,41)(3,15,105,42)(4,16,106,43)(5,66,93,40)(6,67,94,37)(7,68,95,38)(8,65,96,39)(9,103,48,76)(10,104,45,73)(11,101,46,74)(12,102,47,75)(17,109,56,82)(18,110,53,83)(19,111,54,84)(20,112,55,81)(21,113,52,78)(22,114,49,79)(23,115,50,80)(24,116,51,77)(25,119,64,92)(26,120,61,89)(27,117,62,90)(28,118,63,91)(29,123,60,88)(30,124,57,85)(31,121,58,86)(32,122,59,87)(33,125,71,98)(34,126,72,99)(35,127,69,100)(36,128,70,97), (1,109,101,80)(2,83,102,116)(3,111,103,78)(4,81,104,114)(5,118,99,85)(6,92,100,121)(7,120,97,87)(8,90,98,123)(9,50,42,17)(10,24,43,53)(11,52,44,19)(12,22,41,55)(13,54,46,21)(14,20,47,49)(15,56,48,23)(16,18,45,51)(25,33,58,65)(26,72,59,40)(27,35,60,67)(28,70,57,38)(29,37,62,69)(30,68,63,36)(31,39,64,71)(32,66,61,34)(73,79,106,112)(74,115,107,82)(75,77,108,110)(76,113,105,84)(86,94,119,127)(88,96,117,125)(89,128,122,95)(91,126,124,93), (1,117,101,88)(2,120,102,87)(3,119,103,86)(4,118,104,85)(5,114,99,81)(6,113,100,84)(7,116,97,83)(8,115,98,82)(9,58,42,25)(10,57,43,28)(11,60,44,27)(12,59,41,26)(13,62,46,29)(14,61,47,32)(15,64,48,31)(16,63,45,30)(17,65,50,33)(18,68,51,36)(19,67,52,35)(20,66,49,34)(21,69,54,37)(22,72,55,40)(23,71,56,39)(24,70,53,38)(73,124,106,91)(74,123,107,90)(75,122,108,89)(76,121,105,92)(77,128,110,95)(78,127,111,94)(79,126,112,93)(80,125,109,96) );
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)], [(1,13,107,44),(2,14,108,41),(3,15,105,42),(4,16,106,43),(5,66,93,40),(6,67,94,37),(7,68,95,38),(8,65,96,39),(9,103,48,76),(10,104,45,73),(11,101,46,74),(12,102,47,75),(17,109,56,82),(18,110,53,83),(19,111,54,84),(20,112,55,81),(21,113,52,78),(22,114,49,79),(23,115,50,80),(24,116,51,77),(25,119,64,92),(26,120,61,89),(27,117,62,90),(28,118,63,91),(29,123,60,88),(30,124,57,85),(31,121,58,86),(32,122,59,87),(33,125,71,98),(34,126,72,99),(35,127,69,100),(36,128,70,97)], [(1,109,101,80),(2,83,102,116),(3,111,103,78),(4,81,104,114),(5,118,99,85),(6,92,100,121),(7,120,97,87),(8,90,98,123),(9,50,42,17),(10,24,43,53),(11,52,44,19),(12,22,41,55),(13,54,46,21),(14,20,47,49),(15,56,48,23),(16,18,45,51),(25,33,58,65),(26,72,59,40),(27,35,60,67),(28,70,57,38),(29,37,62,69),(30,68,63,36),(31,39,64,71),(32,66,61,34),(73,79,106,112),(74,115,107,82),(75,77,108,110),(76,113,105,84),(86,94,119,127),(88,96,117,125),(89,128,122,95),(91,126,124,93)], [(1,117,101,88),(2,120,102,87),(3,119,103,86),(4,118,104,85),(5,114,99,81),(6,113,100,84),(7,116,97,83),(8,115,98,82),(9,58,42,25),(10,57,43,28),(11,60,44,27),(12,59,41,26),(13,62,46,29),(14,61,47,32),(15,64,48,31),(16,63,45,30),(17,65,50,33),(18,68,51,36),(19,67,52,35),(20,66,49,34),(21,69,54,37),(22,72,55,40),(23,71,56,39),(24,70,53,38),(73,124,106,91),(74,123,107,90),(75,122,108,89),(76,121,105,92),(77,128,110,95),(78,127,111,94),(79,126,112,93),(80,125,109,96)])
Matrix representation ►G ⊆ GL8(𝔽5)
4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 |
0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 | 3 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |
2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 | 2 | 4 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |
4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 3 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 3 | 2 | 1 | 4 |
0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
0 | 0 | 0 | 0 | 1 | 3 | 3 | 1 |
0 | 0 | 0 | 0 | 4 | 4 | 2 | 4 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,2,4,1,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,3],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,4,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,3],[4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,3,0,1,4,0,0,0,0,2,0,3,4,0,0,0,0,1,3,3,2,0,0,0,0,4,3,1,4] >;
32 conjugacy classes
class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4X |
order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | - | + | - | |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
kernel | C42⋊11Q8 | C4×C4⋊C4 | C42⋊8C4 | C42⋊5C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.78C23 | C23.81C23 | C23.83C23 | C42 | C2×C4 | C22 | C22 |
# reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{11}Q_8
% in TeX
G:=Group("C4^2:11Q8");
// GroupNames label
G:=SmallGroup(128,1398);
// by ID
G=gap.SmallGroup(128,1398);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,352,185,136]);
// 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*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations