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G = Q8.M4(2)  order 128 = 27

2nd non-split extension by Q8 of M4(2) acting via M4(2)/C8=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q8.2M4(2), C42.644C23, Q8⋊C8.8C2, C82C8.6C2, C8⋊C8.2C2, C2.8(C8○D8), (C2×C8).309D4, (C2×Q16).7C4, (C8×Q8).15C2, C2.D8.12C4, C4.36(C8○D4), Q8⋊C4.8C4, (C4×Q16).12C2, C84Q8.12C2, C2.14(C89D4), C4⋊C8.278C22, (C4×C8).316C22, C22.135(C4×D4), C4.30(C2×M4(2)), (C4×Q8).12C22, C2.4(Q16⋊C4), C4.138(C8.C22), C4⋊C4.138(C2×C4), (C2×C8).105(C2×C4), (C2×C4).1480(C2×D4), (C2×Q8).137(C2×C4), (C2×C4).505(C4○D4), (C2×C4).336(C22×C4), SmallGroup(128,319)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q8.M4(2)
C1C2C22C2×C4C42C4×C8C8×Q8 — Q8.M4(2)
C1C2C2×C4 — Q8.M4(2)
C1C2×C4C4×C8 — Q8.M4(2)
C1C22C22C42 — Q8.M4(2)

Generators and relations for Q8.M4(2)
 G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a-1b, dcd-1=a2c5 >

Subgroups: 120 in 76 conjugacy classes, 42 normal (40 characteristic)
C1, C2 [×3], C4 [×4], C4 [×6], C22, C8 [×8], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×3], C42, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×3], Q16 [×2], C2×Q8 [×2], C4×C8 [×3], C4×C8, C8⋊C4, Q8⋊C4 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C8⋊C8, Q8⋊C8 [×2], C82C8, C4×Q16, C8×Q8, C84Q8, Q8.M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8.C22 [×2], C89D4, Q16⋊C4, C8○D8, Q8.M4(2)

Smallest permutation representation of Q8.M4(2)
Regular action on 128 points
Generators in S128
(1 44 97 79)(2 80 98 45)(3 46 99 73)(4 74 100 47)(5 48 101 75)(6 76 102 41)(7 42 103 77)(8 78 104 43)(9 123 68 36)(10 37 69 124)(11 125 70 38)(12 39 71 126)(13 127 72 40)(14 33 65 128)(15 121 66 34)(16 35 67 122)(17 86 112 51)(18 52 105 87)(19 88 106 53)(20 54 107 81)(21 82 108 55)(22 56 109 83)(23 84 110 49)(24 50 111 85)(25 59 120 94)(26 95 113 60)(27 61 114 96)(28 89 115 62)(29 63 116 90)(30 91 117 64)(31 57 118 92)(32 93 119 58)
(1 117 97 30)(2 57 98 92)(3 119 99 32)(4 59 100 94)(5 113 101 26)(6 61 102 96)(7 115 103 28)(8 63 104 90)(9 85 68 50)(10 17 69 112)(11 87 70 52)(12 19 71 106)(13 81 72 54)(14 21 65 108)(15 83 66 56)(16 23 67 110)(18 125 105 38)(20 127 107 40)(22 121 109 34)(24 123 111 36)(25 47 120 74)(27 41 114 76)(29 43 116 78)(31 45 118 80)(33 55 128 82)(35 49 122 84)(37 51 124 86)(39 53 126 88)(42 89 77 62)(44 91 79 64)(46 93 73 58)(48 95 75 60)
(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 19 97 106)(2 111 98 24)(3 21 99 108)(4 105 100 18)(5 23 101 110)(6 107 102 20)(7 17 103 112)(8 109 104 22)(9 92 68 57)(10 62 69 89)(11 94 70 59)(12 64 71 91)(13 96 72 61)(14 58 65 93)(15 90 66 63)(16 60 67 95)(25 125 120 38)(26 35 113 122)(27 127 114 40)(28 37 115 124)(29 121 116 34)(30 39 117 126)(31 123 118 36)(32 33 119 128)(41 81 76 54)(42 51 77 86)(43 83 78 56)(44 53 79 88)(45 85 80 50)(46 55 73 82)(47 87 74 52)(48 49 75 84)

G:=sub<Sym(128)| (1,44,97,79)(2,80,98,45)(3,46,99,73)(4,74,100,47)(5,48,101,75)(6,76,102,41)(7,42,103,77)(8,78,104,43)(9,123,68,36)(10,37,69,124)(11,125,70,38)(12,39,71,126)(13,127,72,40)(14,33,65,128)(15,121,66,34)(16,35,67,122)(17,86,112,51)(18,52,105,87)(19,88,106,53)(20,54,107,81)(21,82,108,55)(22,56,109,83)(23,84,110,49)(24,50,111,85)(25,59,120,94)(26,95,113,60)(27,61,114,96)(28,89,115,62)(29,63,116,90)(30,91,117,64)(31,57,118,92)(32,93,119,58), (1,117,97,30)(2,57,98,92)(3,119,99,32)(4,59,100,94)(5,113,101,26)(6,61,102,96)(7,115,103,28)(8,63,104,90)(9,85,68,50)(10,17,69,112)(11,87,70,52)(12,19,71,106)(13,81,72,54)(14,21,65,108)(15,83,66,56)(16,23,67,110)(18,125,105,38)(20,127,107,40)(22,121,109,34)(24,123,111,36)(25,47,120,74)(27,41,114,76)(29,43,116,78)(31,45,118,80)(33,55,128,82)(35,49,122,84)(37,51,124,86)(39,53,126,88)(42,89,77,62)(44,91,79,64)(46,93,73,58)(48,95,75,60), (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), (1,19,97,106)(2,111,98,24)(3,21,99,108)(4,105,100,18)(5,23,101,110)(6,107,102,20)(7,17,103,112)(8,109,104,22)(9,92,68,57)(10,62,69,89)(11,94,70,59)(12,64,71,91)(13,96,72,61)(14,58,65,93)(15,90,66,63)(16,60,67,95)(25,125,120,38)(26,35,113,122)(27,127,114,40)(28,37,115,124)(29,121,116,34)(30,39,117,126)(31,123,118,36)(32,33,119,128)(41,81,76,54)(42,51,77,86)(43,83,78,56)(44,53,79,88)(45,85,80,50)(46,55,73,82)(47,87,74,52)(48,49,75,84)>;

G:=Group( (1,44,97,79)(2,80,98,45)(3,46,99,73)(4,74,100,47)(5,48,101,75)(6,76,102,41)(7,42,103,77)(8,78,104,43)(9,123,68,36)(10,37,69,124)(11,125,70,38)(12,39,71,126)(13,127,72,40)(14,33,65,128)(15,121,66,34)(16,35,67,122)(17,86,112,51)(18,52,105,87)(19,88,106,53)(20,54,107,81)(21,82,108,55)(22,56,109,83)(23,84,110,49)(24,50,111,85)(25,59,120,94)(26,95,113,60)(27,61,114,96)(28,89,115,62)(29,63,116,90)(30,91,117,64)(31,57,118,92)(32,93,119,58), (1,117,97,30)(2,57,98,92)(3,119,99,32)(4,59,100,94)(5,113,101,26)(6,61,102,96)(7,115,103,28)(8,63,104,90)(9,85,68,50)(10,17,69,112)(11,87,70,52)(12,19,71,106)(13,81,72,54)(14,21,65,108)(15,83,66,56)(16,23,67,110)(18,125,105,38)(20,127,107,40)(22,121,109,34)(24,123,111,36)(25,47,120,74)(27,41,114,76)(29,43,116,78)(31,45,118,80)(33,55,128,82)(35,49,122,84)(37,51,124,86)(39,53,126,88)(42,89,77,62)(44,91,79,64)(46,93,73,58)(48,95,75,60), (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), (1,19,97,106)(2,111,98,24)(3,21,99,108)(4,105,100,18)(5,23,101,110)(6,107,102,20)(7,17,103,112)(8,109,104,22)(9,92,68,57)(10,62,69,89)(11,94,70,59)(12,64,71,91)(13,96,72,61)(14,58,65,93)(15,90,66,63)(16,60,67,95)(25,125,120,38)(26,35,113,122)(27,127,114,40)(28,37,115,124)(29,121,116,34)(30,39,117,126)(31,123,118,36)(32,33,119,128)(41,81,76,54)(42,51,77,86)(43,83,78,56)(44,53,79,88)(45,85,80,50)(46,55,73,82)(47,87,74,52)(48,49,75,84) );

G=PermutationGroup([(1,44,97,79),(2,80,98,45),(3,46,99,73),(4,74,100,47),(5,48,101,75),(6,76,102,41),(7,42,103,77),(8,78,104,43),(9,123,68,36),(10,37,69,124),(11,125,70,38),(12,39,71,126),(13,127,72,40),(14,33,65,128),(15,121,66,34),(16,35,67,122),(17,86,112,51),(18,52,105,87),(19,88,106,53),(20,54,107,81),(21,82,108,55),(22,56,109,83),(23,84,110,49),(24,50,111,85),(25,59,120,94),(26,95,113,60),(27,61,114,96),(28,89,115,62),(29,63,116,90),(30,91,117,64),(31,57,118,92),(32,93,119,58)], [(1,117,97,30),(2,57,98,92),(3,119,99,32),(4,59,100,94),(5,113,101,26),(6,61,102,96),(7,115,103,28),(8,63,104,90),(9,85,68,50),(10,17,69,112),(11,87,70,52),(12,19,71,106),(13,81,72,54),(14,21,65,108),(15,83,66,56),(16,23,67,110),(18,125,105,38),(20,127,107,40),(22,121,109,34),(24,123,111,36),(25,47,120,74),(27,41,114,76),(29,43,116,78),(31,45,118,80),(33,55,128,82),(35,49,122,84),(37,51,124,86),(39,53,126,88),(42,89,77,62),(44,91,79,64),(46,93,73,58),(48,95,75,60)], [(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)], [(1,19,97,106),(2,111,98,24),(3,21,99,108),(4,105,100,18),(5,23,101,110),(6,107,102,20),(7,17,103,112),(8,109,104,22),(9,92,68,57),(10,62,69,89),(11,94,70,59),(12,64,71,91),(13,96,72,61),(14,58,65,93),(15,90,66,63),(16,60,67,95),(25,125,120,38),(26,35,113,122),(27,127,114,40),(28,37,115,124),(29,121,116,34),(30,39,117,126),(31,123,118,36),(32,33,119,128),(41,81,76,54),(42,51,77,86),(43,83,78,56),(44,53,79,88),(45,85,80,50),(46,55,73,82),(47,87,74,52),(48,49,75,84)])

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E···8R8S8T
order12224444444444444488888···888
size11111111222244448822224···488

38 irreducible representations

dim1111111111222224
type++++++++-
imageC1C2C2C2C2C2C2C4C4C4D4C4○D4M4(2)C8○D4C8○D8C8.C22
kernelQ8.M4(2)C8⋊C8Q8⋊C8C82C8C4×Q16C8×Q8C84Q8Q8⋊C4C2.D8C2×Q16C2×C8C2×C4Q8C4C2C4
# reps1121111422224482

Matrix representation of Q8.M4(2) in GL4(𝔽17) generated by

11500
11600
00160
00016
,
0700
12000
0009
0020
,
91600
0800
0080
0009
,
13800
0400
0002
0090
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,16,0,0,0,0,16],[0,12,0,0,7,0,0,0,0,0,0,2,0,0,9,0],[9,0,0,0,16,8,0,0,0,0,8,0,0,0,0,9],[13,0,0,0,8,4,0,0,0,0,0,9,0,0,2,0] >;

Q8.M4(2) in GAP, Magma, Sage, TeX

Q_8.M_4(2)
% in TeX

G:=Group("Q8.M4(2)");
// GroupNames label

G:=SmallGroup(128,319);
// by ID

G=gap.SmallGroup(128,319);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,2102,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^5>;
// generators/relations

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