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G = (C2×C4)⋊2Q16order 128 = 27

1st semidirect product of C2×C4 and Q16 acting via Q16/C4=C22

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

Aliases: (C2×C4)⋊2Q16, (C2×C8).42D4, (C2×Q8).85D4, C4.12C22≀C2, C4.48(C4⋊D4), C2.8(C4⋊Q16), C23.904(C2×D4), (C22×C4).310D4, (C22×Q16).3C2, C22.53(C2×Q16), C2.12(C42Q16), C2.15(C8.2D4), C22.206C22≀C2, C22.77(C41D4), C2.21(C232D4), (C22×C8).109C22, (C2×C42).352C22, C2.19(C22⋊Q16), (C22×Q8).57C22, C22.225(C4⋊D4), (C22×C4).1438C23, C22.120(C8.C22), C22.7C42.23C2, C23.67C23.12C2, (C2×C4⋊Q8).18C2, (C2×C4).747(C2×D4), (C2×C4).877(C4○D4), (C2×C4⋊C4).111C22, (C2×Q8⋊C4).20C2, SmallGroup(128,748)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4)⋊2Q16
C1C2C22C2×C4C22×C4C22×Q8C2×C4⋊Q8 — (C2×C4)⋊2Q16
C1C2C22×C4 — (C2×C4)⋊2Q16
C1C23C2×C42 — (C2×C4)⋊2Q16
C1C2C2C22×C4 — (C2×C4)⋊2Q16

Generators and relations for (C2×C4)⋊2Q16
 G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 360 in 185 conjugacy classes, 58 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×19], Q8 [×18], C23, C42 [×2], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], Q16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×15], C2.C42 [×2], Q8⋊C4 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊Q8 [×4], C22×C8 [×2], C2×Q16 [×12], C22×Q8, C22×Q8 [×2], C22.7C42, C23.67C23, C2×Q8⋊C4 [×2], C2×C4⋊Q8, C22×Q16 [×2], (C2×C4)⋊2Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, Q16 [×4], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×Q16 [×2], C8.C22 [×2], C232D4, C22⋊Q16 [×2], C42Q16 [×2], C4⋊Q16, C8.2D4, (C2×C4)⋊2Q16

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111222224
type+++++++++--
imageC1C2C2C2C2C2D4D4D4Q16C4○D4C8.C22
kernel(C2×C4)⋊2Q16C22.7C42C23.67C23C2×Q8⋊C4C2×C4⋊Q8C22×Q16C2×C8C22×C4C2×Q8C2×C4C2×C4C22
# reps111212426822

Matrix representation of (C2×C4)⋊2Q16 in GL6(𝔽17)

1600000
0160000
0016000
0001600
000010
000001
,
010000
1600000
004000
0001300
0000016
000010
,
3140000
330000
004000
0001300
000001
000010
,
10160000
1670000
000100
001000
000001
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

(C2×C4)⋊2Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_2Q_{16}
% in TeX

G:=Group("(C2xC4):2Q16");
// GroupNames label

G:=SmallGroup(128,748);
// by ID

G=gap.SmallGroup(128,748);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,352,2019,1018,521,248,2804,1411,718,172]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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