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G = (C2×Q16)⋊10C4order 128 = 27

6th semidirect product of C2×Q16 and C4 acting via C4/C2=C2

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

Aliases: C4.20(C4×D4), (C2×Q16)⋊10C4, (C2×C8).206D4, C8.2(C22⋊C4), C4.84(C4⋊D4), C2.4(C8.2D4), C4.4(C4.4D4), C2.5(C8.D4), C23.804(C2×D4), C22.184(C4×D4), (C22×C4).134D4, (C22×Q16).11C2, C2.12(Q16⋊C4), C22.40(C41D4), (C2×C42).322C22, (C22×C8).406C22, (C22×Q8).40C22, C22.146(C4⋊D4), (C22×C4).1411C23, C22.85(C8.C22), C23.67C23.11C2, C2.23(C24.3C22), (C2×C8).71(C2×C4), (C2×C8⋊C4).9C2, (C2×C4.Q8).8C2, C4.39(C2×C22⋊C4), (C2×Q8).96(C2×C4), (C2×C4).1356(C2×D4), (C2×C4⋊C4).92C22, (C2×C4).602(C4○D4), (C2×C4).425(C22×C4), (C2×Q8⋊C4).35C2, SmallGroup(128,703)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×Q16)⋊10C4
C1C2C22C2×C4C22×C4C22×C8C2×C8⋊C4 — (C2×Q16)⋊10C4
C1C2C2×C4 — (C2×Q16)⋊10C4
C1C23C2×C42 — (C2×Q16)⋊10C4
C1C2C2C22×C4 — (C2×Q16)⋊10C4

Generators and relations for (C2×Q16)⋊10C4
 G = < a,b,c,d | a2=b8=d4=1, c2=b4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b3, dcd-1=ab4c >

Subgroups: 292 in 152 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×20], Q8 [×12], C23, C42 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C2.C42 [×4], C8⋊C4 [×2], Q8⋊C4 [×4], C4.Q8 [×2], C2×C42, C2×C4⋊C4 [×2], C22×C8 [×2], C2×Q16 [×4], C2×Q16 [×4], C22×Q8 [×2], C23.67C23 [×2], C2×C8⋊C4, C2×Q8⋊C4 [×2], C2×C4.Q8, C22×Q16, (C2×Q16)⋊10C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C8.C22 [×4], C24.3C22, Q16⋊C4 [×2], C8.D4 [×2], C8.2D4 [×2], (C2×Q16)⋊10C4

Smallest permutation representation of (C2×Q16)⋊10C4
Regular action on 128 points
Generators in S128
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 127)(26 128)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 85)(66 86)(67 87)(68 88)(69 81)(70 82)(71 83)(72 84)(89 109)(90 110)(91 111)(92 112)(93 105)(94 106)(95 107)(96 108)(97 118)(98 119)(99 120)(100 113)(101 114)(102 115)(103 116)(104 117)
(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)
(1 117 5 113)(2 116 6 120)(3 115 7 119)(4 114 8 118)(9 123 13 127)(10 122 14 126)(11 121 15 125)(12 128 16 124)(17 104 21 100)(18 103 22 99)(19 102 23 98)(20 101 24 97)(25 41 29 45)(26 48 30 44)(27 47 31 43)(28 46 32 42)(33 88 37 84)(34 87 38 83)(35 86 39 82)(36 85 40 81)(49 72 53 68)(50 71 54 67)(51 70 55 66)(52 69 56 65)(57 112 61 108)(58 111 62 107)(59 110 63 106)(60 109 64 105)(73 91 77 95)(74 90 78 94)(75 89 79 93)(76 96 80 92)
(1 86 46 111)(2 81 47 106)(3 84 48 109)(4 87 41 112)(5 82 42 107)(6 85 43 110)(7 88 44 105)(8 83 45 108)(9 92 20 67)(10 95 21 70)(11 90 22 65)(12 93 23 68)(13 96 24 71)(14 91 17 66)(15 94 18 69)(16 89 19 72)(25 76 118 50)(26 79 119 53)(27 74 120 56)(28 77 113 51)(29 80 114 54)(30 75 115 49)(31 78 116 52)(32 73 117 55)(33 128 64 98)(34 123 57 101)(35 126 58 104)(36 121 59 99)(37 124 60 102)(38 127 61 97)(39 122 62 100)(40 125 63 103)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim11111112224
type++++++++-
imageC1C2C2C2C2C2C4D4D4C4○D4C8.C22
kernel(C2×Q16)⋊10C4C23.67C23C2×C8⋊C4C2×Q8⋊C4C2×C4.Q8C22×Q16C2×Q16C2×C8C22×C4C2×C4C22
# reps12121186244

Matrix representation of (C2×Q16)⋊10C4 in GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
00010000
001600000
000001300
000016000
000040415
0000891013
,
411000000
1113000000
000160000
001600000
00003047
0000130162
00001500
0000152814
,
716000000
1610000000
00100000
000160000
00000010
000040415
000016000
000015920

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,4,8,0,0,0,0,13,0,0,9,0,0,0,0,0,0,4,10,0,0,0,0,0,0,15,13],[4,11,0,0,0,0,0,0,11,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,13,1,15,0,0,0,0,0,0,5,2,0,0,0,0,4,16,0,8,0,0,0,0,7,2,0,14],[7,16,0,0,0,0,0,0,16,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,16,15,0,0,0,0,0,0,0,9,0,0,0,0,1,4,0,2,0,0,0,0,0,15,0,0] >;

(C2×Q16)⋊10C4 in GAP, Magma, Sage, TeX

(C_2\times Q_{16})\rtimes_{10}C_4
% in TeX

G:=Group("(C2xQ16):10C4");
// GroupNames label

G:=SmallGroup(128,703);
// by ID

G=gap.SmallGroup(128,703);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^3,d*c*d^-1=a*b^4*c>;
// generators/relations

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