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G = (C2×C4).23Q16order 128 = 27

20th non-split extension by C2×C4 of Q16 acting via Q16/C4=C22

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

Aliases: C4⋊C4.6Q8, (C2×C4).23Q16, (C2×C4).46SD16, C23.946(C2×D4), (C22×C4).329D4, C22.65(C2×Q16), C429C4.21C2, C2.11(C4.Q16), C2.11(D42Q8), (C22×C8).91C22, C2.8(C4.SD16), C4.26(C42.C2), C4.23(C422C2), C22.4Q16.31C2, (C2×C42).397C22, C22.111(C2×SD16), C22.161(C8⋊C22), (C22×C4).1480C23, C22.98(C4.4D4), C22.123(C22⋊Q8), C2.11(C23.46D4), C2.11(C23.48D4), C22.7C42.16C2, C23.65C23.27C2, C2.8(C42.29C22), C2.6(C23.83C23), C22.133(C22.D4), (C2×C4).297(C2×Q8), (C2×C4).786(C4○D4), (C2×C4⋊C4).169C22, SmallGroup(128,832)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4).23Q16
C1C2C4C2×C4C22×C4C2×C4⋊C4C22.4Q16 — (C2×C4).23Q16
C1C2C22×C4 — (C2×C4).23Q16
C1C23C2×C42 — (C2×C4).23Q16
C1C2C2C22×C4 — (C2×C4).23Q16

Generators and relations for (C2×C4).23Q16
 G = < a,b,c,d | a2=b4=c8=1, d2=b2c4, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd-1=ac-1 >

Subgroups: 224 in 107 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], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C22×C8 [×2], C22.7C42, C22.4Q16 [×4], C429C4, C23.65C23, (C2×C4).23Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C4○D4 [×5], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], C2×SD16, C2×Q16, C8⋊C22 [×2], C23.83C23, D42Q8, C4.Q16, C23.46D4, C23.48D4, C4.SD16, C42.29C22, (C2×C4).23Q16

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111222224
type+++++-+-+
imageC1C2C2C2C2Q8D4SD16Q16C4○D4C8⋊C22
kernel(C2×C4).23Q16C22.7C42C22.4Q16C429C4C23.65C23C4⋊C4C22×C4C2×C4C2×C4C2×C4C22
# reps114112244102

Matrix representation of (C2×C4).23Q16 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
5120000
12120000
008300
001900
0000162
0000161
,
1430000
330000
0013800
0013400
000007
000057
,
010000
1600000
004000
000400
0000160
0000161

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,8,1,0,0,0,0,3,9,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[14,3,0,0,0,0,3,3,0,0,0,0,0,0,13,13,0,0,0,0,8,4,0,0,0,0,0,0,0,5,0,0,0,0,7,7],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,0,0,0,0,1] >;

(C2×C4).23Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{23}Q_{16}
% in TeX

G:=Group("(C2xC4).23Q16");
// GroupNames label

G:=SmallGroup(128,832);
// by ID

G=gap.SmallGroup(128,832);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,624,422,387,58,2028,1027,124]);
// Polycyclic

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

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