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G = (C2×C8).Q8order 128 = 27

30th non-split extension by C2×C8 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — (C2×C8).Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — (C2×C8).Q8
 Lower central C1 — C23 — (C2×C8).Q8
 Upper central C1 — C22×C4 — (C2×C8).Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).Q8

Generators and relations for (C2×C8).Q8
G = < a,b,c,d | a2=b8=1, c4=b4, d2=b4c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=ab5, dcd-1=b6c3 >

Subgroups: 180 in 114 conjugacy classes, 60 normal (52 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C2.C42, C8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C22.7C42, C4×C4⋊C4, C2×C8⋊C4, C2×C4⋊C8, (C2×C8).Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×M4(2), C8○D4, C23.63C23, C42.6C4, C42.7C22, C89D4, C86D4, C84Q8, (C2×C8).Q8

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 M4(2) C4○D4 C8○D4 kernel (C2×C8).Q8 C22.7C42 C4×C4⋊C4 C2×C8⋊C4 C2×C4⋊C8 C2.C42 C2×C4⋊C4 C2×C8 C2×C8 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 4 4 2 2 8 8 8

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

 16 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 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 5 8 0 0 0 0 8 12 0 0 0 0 0 0 13 9 0 0 0 0 4 4
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 2 3 0 0 0 0 3 15 0 0 0 0 0 0 1 2 0 0 0 0 16 16
,
 6 1 0 0 0 0 16 11 0 0 0 0 0 0 12 9 0 0 0 0 9 5 0 0 0 0 0 0 13 9 0 0 0 0 0 4

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,5,8,0,0,0,0,8,12,0,0,0,0,0,0,13,4,0,0,0,0,9,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,2,3,0,0,0,0,3,15,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[6,16,0,0,0,0,1,11,0,0,0,0,0,0,12,9,0,0,0,0,9,5,0,0,0,0,0,0,13,0,0,0,0,0,9,4] >;

(C2×C8).Q8 in GAP, Magma, Sage, TeX

(C_2\times C_8).Q_8
% in TeX

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

G:=SmallGroup(128,649);
// by ID

G=gap.SmallGroup(128,649);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,58,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=a*b^5,d*c*d^-1=b^6*c^3>;
// generators/relations

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