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G = (C2×C4)⋊9Q16order 128 = 27

1st semidirect product of C2×C4 and Q16 acting via Q16/Q8=C2

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

Aliases: (C2×C4)⋊9Q16, (C2×Q16)⋊6C4, C4.82(C4×D4), C2.7(C4×Q16), C4⋊C4.311D4, C4.2(C4⋊D4), (C2×Q8).162D4, C2.2(C42Q16), (C22×C4).688D4, C23.769(C2×D4), C22.152(C4×D4), Q8.2(C22⋊C4), (C22×Q16).1C2, C22.32(C2×Q16), C2.7(Q16⋊C4), C22.94C22≀C2, C2.7(D4.7D4), C22.56(C4○D8), (C22×C8).36C22, C2.5(C22⋊Q16), C2.2(Q8.D4), C22.4Q16.34C2, (C2×C42).278C22, C22.115(C4⋊D4), (C22×C4).1365C23, C4.64(C22.D4), C22.66(C8.C22), (C22×Q8).387C22, C2.17(C23.23D4), C22.7C42.19C2, C23.67C23.5C2, (C2×C4×Q8).15C2, (C2×C8).37(C2×C4), (C2×C4).997(C2×D4), C4.12(C2×C22⋊C4), (C2×Q8).64(C2×C4), (C2×Q8⋊C4).3C2, (C2×C4).562(C4○D4), (C2×C4⋊C4).766C22, (C2×C4).383(C22×C4), SmallGroup(128,610)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×C4)⋊9Q16
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — (C2×C4)⋊9Q16
C1C2C2×C4 — (C2×C4)⋊9Q16
C1C23C2×C42 — (C2×C4)⋊9Q16
C1C2C2C22×C4 — (C2×C4)⋊9Q16

Generators and relations for (C2×C4)⋊9Q16
 G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=ab-1, bd=db, dcd-1=c-1 >

Subgroups: 308 in 169 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×12], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×22], Q8 [×4], Q8 [×12], C23, C42 [×6], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], Q16 [×8], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C2×Q8 [×8], C2.C42 [×2], Q8⋊C4 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×Q8 [×4], C22×C8 [×2], C2×Q16 [×4], C2×Q16 [×4], C22×Q8 [×2], C22.7C42, C22.4Q16, C23.67C23, C2×Q8⋊C4 [×2], C2×C4×Q8, C22×Q16, (C2×C4)⋊9Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], Q16 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×Q16, C4○D8, C8.C22 [×2], C23.23D4, C4×Q16, Q16⋊C4, C22⋊Q16, D4.7D4, C42Q16, Q8.D4, (C2×C4)⋊9Q16

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

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim111111112222224
type++++++++++--
imageC1C2C2C2C2C2C2C4D4D4D4Q16C4○D4C4○D8C8.C22
kernel(C2×C4)⋊9Q16C22.7C42C22.4Q16C23.67C23C2×Q8⋊C4C2×C4×Q8C22×Q16C2×Q16C4⋊C4C22×C4C2×Q8C2×C4C2×C4C22C22
# reps111121182244442

Matrix representation of (C2×C4)⋊9Q16 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
0160000
100000
004000
000400
0000160
0000016
,
0160000
1600000
00101100
0014700
000033
0000143
,
1600000
0160000
00161600
000100
000017
0000716

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,10,14,0,0,0,0,11,7,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,0,0,0,1,7,0,0,0,0,7,16] >;

(C2×C4)⋊9Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_9Q_{16}
% in TeX

G:=Group("(C2xC4):9Q16");
// GroupNames label

G:=SmallGroup(128,610);
// by ID

G=gap.SmallGroup(128,610);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic

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

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