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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

Aliases: (C2×C8).30Q8, (C2×C8).197D4, C2.6(C84Q8), C22.40(C4×Q8), C2.18(C89D4), C2.14(C86D4), C22.155(C4×D4), (C2×C4).19M4(2), (C2×C42).5C22, C22.53(C8○D4), C4.115(C22⋊Q8), (C22×C8).41C22, C4.37(C42.C2), C4.47(C422C2), C2.C42.22C4, C2.8(C42.6C4), C23.314(C22×C4), C22.67(C2×M4(2)), (C22×C4).1632C23, C22.82(C42⋊C2), C4.138(C22.D4), C22.7C42.44C2, C2.10(C42.7C22), C2.8(C23.63C23), (C2×C4⋊C8).55C2, (C2×C4⋊C4).57C4, (C4×C4⋊C4).13C2, (C2×C4).344(C2×Q8), (C2×C8⋊C4).30C2, (C2×C4).1533(C2×D4), (C2×C4).938(C4○D4), (C22×C4).122(C2×C4), SmallGroup(128,649)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — (C2×C8).Q8
C1C2C4C2×C4C22×C4C2×C42C2×C8⋊C4 — (C2×C8).Q8
C1C23 — (C2×C8).Q8
C1C22×C4 — (C2×C8).Q8
C1C2C2C22×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 [×7], C4 [×4], C4 [×8], C22 [×7], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], C23, C42 [×6], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×10], C22×C4 [×7], C2.C42 [×2], C8⋊C4 [×2], C4⋊C8 [×2], C2×C42 [×3], C2×C4⋊C4 [×2], C22×C8 [×4], C22.7C42 [×4], C4×C4⋊C4, C2×C8⋊C4, C2×C4⋊C8, (C2×C8).Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, M4(2) [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×M4(2) [×2], C8○D4 [×2], C23.63C23, C42.6C4, C42.7C22, C89D4, C86D4, C84Q8 [×2], (C2×C8).Q8

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

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4H4I···4T8A···8P
order12···24···44···48···8
size11···11···14···44···4

44 irreducible representations

dim111111122222
type++++++-
imageC1C2C2C2C2C4C4D4Q8M4(2)C4○D4C8○D4
kernel(C2×C8).Q8C22.7C42C4×C4⋊C4C2×C8⋊C4C2×C4⋊C8C2.C42C2×C4⋊C4C2×C8C2×C8C2×C4C2×C4C22
# reps141114422888

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

1600000
0160000
0016000
0001600
0000160
0000016
,
0160000
1600000
005800
0081200
0000139
000044
,
1600000
0160000
002300
0031500
000012
00001616
,
610000
16110000
0012900
009500
0000139
000004

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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