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G = (C2×C8).52D4order 128 = 27

20th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).52D4, (C2×Q8).9Q8, C4⋊C4.109D4, (C2×C4).18Q16, C2.21(C8⋊D4), C2.8(C4.Q16), (C22×C4).157D4, C23.925(C2×D4), C2.11(Q8.Q8), C4.38(C22⋊Q8), C22.59(C2×Q16), C4.156(C4⋊D4), C2.15(C42Q16), (C22×C8).83C22, C4.36(C422C2), C2.21(D4.2D4), C2.15(C8.18D4), C22.119(C4○D8), C22.4Q16.24C2, (C2×C42).379C22, C2.9(C23.Q8), (C22×Q8).76C22, C22.246(C4⋊D4), C22.147(C8⋊C22), (C22×C4).1459C23, C22.112(C22⋊Q8), C22.136(C8.C22), C23.67C23.18C2, C23.65C23.19C2, (C2×C4⋊C8).38C2, (C2×C4).286(C2×Q8), (C2×C2.D8).14C2, (C2×C4).1058(C2×D4), (C2×C4).623(C4○D4), (C2×C4⋊C4).144C22, (C2×Q8⋊C4).15C2, SmallGroup(128,800)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).52D4
C1C2C4C2×C4C22×C4C22×Q8C2×Q8⋊C4 — (C2×C8).52D4
C1C2C22×C4 — (C2×C8).52D4
C1C23C2×C42 — (C2×C8).52D4
C1C2C2C22×C4 — (C2×C8).52D4

Generators and relations for (C2×C8).52D4
 G = < a,b,c,d | a2=b8=c4=1, d2=b2, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=b6c-1 >

Subgroups: 248 in 124 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×19], Q8 [×6], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], Q8⋊C4 [×4], C4⋊C8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C2.D8, (C2×C8).52D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, Q16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, C23.Q8, C42Q16, D4.2D4, C8.18D4, C8⋊D4, C4.Q16, Q8.Q8, (C2×C8).52D4

Smallest permutation representation of (C2×C8).52D4
Regular action on 128 points
Generators in S128
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 127)(34 128)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 112)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(81 93)(82 94)(83 95)(84 96)(85 89)(86 90)(87 91)(88 92)
(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)
(1 121 23 94)(2 128 24 93)(3 127 17 92)(4 126 18 91)(5 125 19 90)(6 124 20 89)(7 123 21 96)(8 122 22 95)(9 58 106 102)(10 57 107 101)(11 64 108 100)(12 63 109 99)(13 62 110 98)(14 61 111 97)(15 60 112 104)(16 59 105 103)(25 48 65 114)(26 47 66 113)(27 46 67 120)(28 45 68 119)(29 44 69 118)(30 43 70 117)(31 42 71 116)(32 41 72 115)(33 80 88 55)(34 79 81 54)(35 78 82 53)(36 77 83 52)(37 76 84 51)(38 75 85 50)(39 74 86 49)(40 73 87 56)
(1 14 3 16 5 10 7 12)(2 115 4 117 6 119 8 113)(9 50 11 52 13 54 15 56)(17 105 19 107 21 109 23 111)(18 43 20 45 22 47 24 41)(25 86 27 88 29 82 31 84)(26 91 28 93 30 95 32 89)(33 69 35 71 37 65 39 67)(34 102 36 104 38 98 40 100)(42 74 44 76 46 78 48 80)(49 118 51 120 53 114 55 116)(57 94 59 96 61 90 63 92)(58 83 60 85 62 87 64 81)(66 126 68 128 70 122 72 124)(73 106 75 108 77 110 79 112)(97 125 99 127 101 121 103 123)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111222222244
type++++++++++--+-
imageC1C2C2C2C2C2C2D4D4D4Q8Q16C4○D4C4○D8C8⋊C22C8.C22
kernel(C2×C8).52D4C22.4Q16C23.65C23C23.67C23C2×Q8⋊C4C2×C4⋊C8C2×C2.D8C4⋊C4C2×C8C22×C4C2×Q8C2×C4C2×C4C22C22C22
# reps1111211222246411

Matrix representation of (C2×C8).52D4 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
330000
1430000
006200
0071100
0000139
000004
,
14140000
1430000
0010900
002700
0000160
000011
,
1250000
12120000
007800
00151000
00001615
000011

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],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,0,0,0,0,13,0,0,0,0,0,9,4],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,10,2,0,0,0,0,9,7,0,0,0,0,0,0,16,1,0,0,0,0,0,1],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,7,15,0,0,0,0,8,10,0,0,0,0,0,0,16,1,0,0,0,0,15,1] >;

(C2×C8).52D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{52}D_4
% in TeX

G:=Group("(C2xC8).52D4");
// GroupNames label

G:=SmallGroup(128,800);
// by ID

G=gap.SmallGroup(128,800);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,512,422,387,352,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^2,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations

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