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G = C2.(C83Q8)  order 128 = 27

2nd central stem extension by C2 of C83Q8

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

Aliases: (C2×C8).23Q8, C4⋊C4.111D4, C2.10(C8⋊Q8), C2.5(C83Q8), (C2×C4).41SD16, (C22×C4).320D4, C23.934(C2×D4), C22.52(C4⋊Q8), C4.12(C22⋊Q8), C429C4.17C2, C2.17(C4⋊SD16), C4.20(C42.C2), C2.17(D4.D4), C22.4Q16.44C2, (C2×C42).385C22, (C22×C8).327C22, C22.106(C2×SD16), C22.257(C4⋊D4), C22.154(C8⋊C22), (C22×C4).1468C23, C2.9(C23.46D4), C2.9(C23.47D4), C22.143(C8.C22), C22.7C42.36C2, C23.65C23.21C2, C2.6(C23.81C23), C22.124(C22.D4), (C2×C4).221(C2×Q8), (C2×C4.Q8).25C2, (C2×C4).1064(C2×D4), (C2×C4).887(C4○D4), (C2×C4⋊C4).155C22, SmallGroup(128,816)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C2.(C83Q8)
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C4.Q8 — C2.(C83Q8)
C1C2C22×C4 — C2.(C83Q8)
C1C23C2×C42 — C2.(C83Q8)
C1C2C2C22×C4 — C2.(C83Q8)

Generators and relations for C2.(C83Q8)
 G = < a,b,c,d | a2=b8=c4=1, d2=c2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=ab3, dcd-1=ac-1 >

Subgroups: 232 in 115 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42, C4.Q8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C22×C8 [×2], C22.7C42, C22.4Q16 [×2], C429C4, C23.65C23, C2×C4.Q8 [×2], C2.(C83Q8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, SD16 [×4], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×SD16 [×2], C8⋊C22, C8.C22, C23.81C23, C4⋊SD16, D4.D4, C23.46D4, C23.47D4, C83Q8, C8⋊Q8, C2.(C83Q8)

Smallest permutation representation of C2.(C83Q8)
Regular action on 128 points
Generators in S128
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(81 127)(82 128)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(105 116)(106 117)(107 118)(108 119)(109 120)(110 113)(111 114)(112 115)
(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 98 90 121)(2 59 91 84)(3 100 92 123)(4 61 93 86)(5 102 94 125)(6 63 95 88)(7 104 96 127)(8 57 89 82)(9 105 37 77)(10 117 38 67)(11 107 39 79)(12 119 40 69)(13 109 33 73)(14 113 34 71)(15 111 35 75)(16 115 36 65)(17 87 53 62)(18 126 54 103)(19 81 55 64)(20 128 56 97)(21 83 49 58)(22 122 50 99)(23 85 51 60)(24 124 52 101)(25 120 48 70)(26 110 41 74)(27 114 42 72)(28 112 43 76)(29 116 44 66)(30 106 45 78)(31 118 46 68)(32 108 47 80)
(1 75 90 111)(2 67 91 117)(3 73 92 109)(4 65 93 115)(5 79 94 107)(6 71 95 113)(7 77 96 105)(8 69 89 119)(9 64 37 81)(10 99 38 122)(11 62 39 87)(12 97 40 128)(13 60 33 85)(14 103 34 126)(15 58 35 83)(16 101 36 124)(17 118 53 68)(18 110 54 74)(19 116 55 66)(20 108 56 80)(21 114 49 72)(22 106 50 78)(23 120 51 70)(24 112 52 76)(25 100 48 123)(26 63 41 88)(27 98 42 121)(28 61 43 86)(29 104 44 127)(30 59 45 84)(31 102 46 125)(32 57 47 82)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111112222244
type+++++++-++-
imageC1C2C2C2C2C2D4Q8D4SD16C4○D4C8⋊C22C8.C22
kernelC2.(C83Q8)C22.7C42C22.4Q16C429C4C23.65C23C2×C4.Q8C4⋊C4C2×C8C22×C4C2×C4C2×C4C22C22
# reps1121122428611

Matrix representation of C2.(C83Q8) in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
1040000
1370000
001000
000100
00001212
0000512
,
1110000
6160000
008200
0010900
0000160
0000016
,
1300000
0130000
0015800
0010200
0000411
00001113

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[1,6,0,0,0,0,11,16,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,15,10,0,0,0,0,8,2,0,0,0,0,0,0,4,11,0,0,0,0,11,13] >;

C2.(C83Q8) in GAP, Magma, Sage, TeX

C_2.(C_8\rtimes_3Q_8)
% in TeX

G:=Group("C2.(C8:3Q8)");
// GroupNames label

G:=SmallGroup(128,816);
// by ID

G=gap.SmallGroup(128,816);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,394,2028,1027,124]);
// Polycyclic

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

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