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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).24Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C2.D8 — (C2×C8).24Q8
 Lower central C1 — C2 — C22×C4 — (C2×C8).24Q8
 Upper central C1 — C23 — C2×C42 — (C2×C8).24Q8
 Jennings C1 — C2 — C2 — C22×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 ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D4 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel (C2×C8).24Q8 C22.7C42 C22.4Q16 C42⋊8C4 C23.65C23 C2×C4.Q8 C2×C2.D8 C4⋊C4 C2×C8 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 2 1 1 1 1 2 4 2 6 8 1 1

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

 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 1 0 0 0 0 16 0 0 0 0 0 0 0 1 9 0 0 0 0 13 16 0 0 0 0 0 0 5 12 0 0 0 0 5 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 8 0 0 0 0 0 1 0 0 0 0 0 0 10 1 0 0 0 0 1 7

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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