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G = (C2×C4).19Q16order 128 = 27

16th non-split extension by C2×C4 of Q16 acting via Q16/C4=C22

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

Aliases: (C2×C4).19Q16, (C2×Q8).108D4, (C2×C4).40SD16, (C22×C4).318D4, C23.927(C2×D4), C22.60(C2×Q16), C429C4.15C2, C2.16(C4⋊SD16), C2.16(C42Q16), (C22×C8).85C22, C2.7(C4.SD16), C4.38(C422C2), C22.4Q16.25C2, (C2×C42).381C22, C22.104(C2×SD16), (C22×Q8).78C22, C22.248(C4⋊D4), C22.149(C8⋊C22), (C22×C4).1461C23, C2.7(C23.47D4), C4.27(C22.D4), C2.7(C23.48D4), C2.7(C23.11D4), C22.94(C4.4D4), C22.138(C8.C22), C22.7C42.13C2, C23.67C23.19C2, C2.10(C42.28C22), C22.117(C22.D4), (C2×C4).1060(C2×D4), (C2×C4).885(C4○D4), (C2×C4⋊C4).146C22, (C2×Q8⋊C4).16C2, SmallGroup(128,804)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4).19Q16
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×Q8⋊C4 — (C2×C4).19Q16
C1C2C22×C4 — (C2×C4).19Q16
C1C23C2×C42 — (C2×C4).19Q16
C1C2C2C22×C4 — (C2×C4).19Q16

Generators and relations for (C2×C4).19Q16
 G = < a,b,c,d | a2=b4=c8=1, d2=ac4, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=ac-1 >

Subgroups: 248 in 119 conjugacy classes, 50 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], Q8 [×6], C23, C42 [×2], C4⋊C4 [×10], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×2], Q8⋊C4 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16 [×2], C429C4, C23.67C23, C2×Q8⋊C4 [×2], (C2×C4).19Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], Q16 [×2], C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C2×SD16, C2×Q16, C8⋊C22, C8.C22, C23.11D4, C4⋊SD16, C42Q16, C23.47D4, C23.48D4, C4.SD16, C42.28C22, (C2×C4).19Q16

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111112222244
type++++++++-+-
imageC1C2C2C2C2C2D4D4SD16Q16C4○D4C8⋊C22C8.C22
kernel(C2×C4).19Q16C22.7C42C22.4Q16C429C4C23.67C23C2×Q8⋊C4C22×C4C2×Q8C2×C4C2×C4C2×C4C22C22
# reps11211222441011

Matrix representation of (C2×C4).19Q16 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
1660000
1110000
001900
00131600
0000115
0000116
,
010000
100000
004000
000400
000007
000057
,
010000
100000
00131500
000400
000006
000030

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,11,0,0,0,0,6,1,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,0,0,0,0,7,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,15,4,0,0,0,0,0,0,0,3,0,0,0,0,6,0] >;

(C2×C4).19Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{19}Q_{16}
% in TeX

G:=Group("(C2xC4).19Q16");
// GroupNames label

G:=SmallGroup(128,804);
// by ID

G=gap.SmallGroup(128,804);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,456,422,387,58,2028,1027,124]);
// Polycyclic

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

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