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G = (C2×C8).24Q8order 128 = 27

24th non-split extension by C2×C8 of Q8 acting via Q8/C2=C22

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

Aliases: (C2×C8).24Q8, C4⋊C4.112D4, C2.11(C8⋊Q8), C23.935(C2×D4), (C22×C4).163D4, C22.53(C4⋊Q8), C4.13(C22⋊Q8), C2.6(C8.5Q8), C428C4.16C2, C4.21(C42.C2), C2.23(D4.2D4), C22.125(C4○D8), C22.4Q16.41C2, (C22×C8).119C22, (C2×C42).386C22, C2.23(Q8.D4), C22.258(C4⋊D4), C22.155(C8⋊C22), (C22×C4).1469C23, C2.13(C23.19D4), C2.13(C23.20D4), C22.144(C8.C22), C22.7C42.28C2, C23.65C23.22C2, C2.7(C23.81C23), C22.125(C22.D4), (C2×C4).222(C2×Q8), (C2×C2.D8).16C2, (C2×C4.Q8).26C2, (C2×C4).1065(C2×D4), (C2×C4).888(C4○D4), (C2×C4⋊C4).156C22, SmallGroup(128,817)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).24Q8
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C2.D8 — (C2×C8).24Q8
C1C2C22×C4 — (C2×C8).24Q8
C1C23C2×C42 — (C2×C8).24Q8
C1C2C2C22×C4 — (C2×C8).24Q8

Generators and relations for (C2×C8).24Q8
 G = < a,b,c,d | a2=b8=c4=1, d2=ab4c2, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=b3, dcd-1=ab4c-1 >

Subgroups: 216 in 107 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×7], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×19], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42 [×3], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C22×C8 [×2], C22.7C42, C22.4Q16 [×2], C428C4, C23.65C23, C2×C4.Q8, C2×C2.D8, (C2×C8).24Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C4○D8 [×2], C8⋊C22, C8.C22, C23.81C23, D4.2D4, Q8.D4, C23.19D4, C23.20D4, C8.5Q8, C8⋊Q8, (C2×C8).24Q8

Smallest permutation representation of (C2×C8).24Q8
Regular action on 128 points
Generators in S128
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 87)(26 88)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 94)(34 95)(35 96)(36 89)(37 90)(38 91)(39 92)(40 93)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 105)(64 106)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(73 104)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)
(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 68 13 106)(2 116 14 59)(3 66 15 112)(4 114 16 57)(5 72 9 110)(6 120 10 63)(7 70 11 108)(8 118 12 61)(17 36 78 28)(18 92 79 85)(19 34 80 26)(20 90 73 83)(21 40 74 32)(22 96 75 81)(23 38 76 30)(24 94 77 87)(25 43 33 100)(27 41 35 98)(29 47 37 104)(31 45 39 102)(42 91 99 84)(44 89 101 82)(46 95 103 88)(48 93 97 86)(49 60 123 117)(50 105 124 67)(51 58 125 115)(52 111 126 65)(53 64 127 113)(54 109 128 71)(55 62 121 119)(56 107 122 69)
(1 104 49 24)(2 99 50 19)(3 102 51 22)(4 97 52 17)(5 100 53 20)(6 103 54 23)(7 98 55 18)(8 101 56 21)(9 43 127 73)(10 46 128 76)(11 41 121 79)(12 44 122 74)(13 47 123 77)(14 42 124 80)(15 45 125 75)(16 48 126 78)(25 72 90 64)(26 67 91 59)(27 70 92 62)(28 65 93 57)(29 68 94 60)(30 71 95 63)(31 66 96 58)(32 69 89 61)(33 110 83 113)(34 105 84 116)(35 108 85 119)(36 111 86 114)(37 106 87 117)(38 109 88 120)(39 112 81 115)(40 107 82 118)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111112222244
type++++++++-++-
imageC1C2C2C2C2C2C2D4Q8D4C4○D4C4○D8C8⋊C22C8.C22
kernel(C2×C8).24Q8C22.7C42C22.4Q16C428C4C23.65C23C2×C4.Q8C2×C2.D8C4⋊C4C2×C8C22×C4C2×C4C22C22C22
# reps11211112426811

Matrix representation of (C2×C8).24Q8 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
010000
1600000
001900
00131600
0000512
000055
,
040000
400000
004000
000400
0000411
00001113
,
400000
0130000
0016800
000100
0000101
000017

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],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,11,0,0,0,0,11,13],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,8,1,0,0,0,0,0,0,10,1,0,0,0,0,1,7] >;

(C2×C8).24Q8 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{24}Q_8
% in TeX

G:=Group("(C2xC8).24Q8");
// GroupNames label

G:=SmallGroup(128,817);
// by ID

G=gap.SmallGroup(128,817);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,64,422,387,58,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a*b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=b^3,d*c*d^-1=a*b^4*c^-1>;
// generators/relations

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