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G = C4.Q810C4order 128 = 27

8th semidirect product of C4.Q8 and C4 acting via C4/C2=C2

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

Aliases: C4.Q810C4, C4⋊C4.13Q8, C4.25(C4×Q8), C2.13(C4×SD16), (C2×C4).88SD16, C2.5(Q8.Q8), C2.3(D42Q8), (C22×C4).693D4, C22.158(C4×D4), C23.778(C2×D4), C22.63(C4○D8), C22.4Q16.9C2, (C22×C8).44C22, C4.13(C42.C2), C22.63(C2×SD16), C4.15(C422C2), C4.24(C42⋊C2), C22.82(C8⋊C22), (C2×C42).293C22, C22.77(C22⋊Q8), C2.13(SD16⋊C4), (C22×C4).1376C23, C2.4(C23.19D4), C2.4(C23.47D4), C22.72(C8.C22), C22.7C42.33C2, C23.65C23.6C2, C22.90(C22.D4), C2.11(C23.63C23), (C4×C4⋊C4).16C2, C4⋊C4.78(C2×C4), (C2×C8).111(C2×C4), (C2×C4).271(C2×Q8), (C2×C4.Q8).17C2, (C2×C4).573(C4○D4), (C2×C4⋊C4).771C22, (C2×C4).394(C22×C4), SmallGroup(128,652)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.Q810C4
C1C2C22C2×C4C22×C4C2×C4⋊C4C4×C4⋊C4 — C4.Q810C4
C1C2C2×C4 — C4.Q810C4
C1C23C2×C42 — C4.Q810C4
C1C2C2C22×C4 — C4.Q810C4

Generators and relations for C4.Q810C4
 G = < a,b,c,d | a4=d4=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, cbc-1=b3, dbd-1=a-1b3, cd=dc >

Subgroups: 220 in 115 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], C23, C42 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4.Q8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C22×C8 [×2], C22.7C42, C22.4Q16 [×3], C4×C4⋊C4, C23.65C23, C2×C4.Q8, C4.Q810C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, SD16 [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.63C23, C4×SD16, SD16⋊C4, D42Q8, Q8.Q8, C23.19D4, C23.47D4, C4.Q810C4

Smallest permutation representation of C4.Q810C4
Regular action on 128 points
Generators in S128
(1 64 5 60)(2 57 6 61)(3 58 7 62)(4 59 8 63)(9 107 13 111)(10 108 14 112)(11 109 15 105)(12 110 16 106)(17 99 21 103)(18 100 22 104)(19 101 23 97)(20 102 24 98)(25 91 29 95)(26 92 30 96)(27 93 31 89)(28 94 32 90)(33 78 37 74)(34 79 38 75)(35 80 39 76)(36 73 40 77)(41 86 45 82)(42 87 46 83)(43 88 47 84)(44 81 48 85)(49 66 53 70)(50 67 54 71)(51 68 55 72)(52 69 56 65)(113 125 117 121)(114 126 118 122)(115 127 119 123)(116 128 120 124)
(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)
(1 107 62 11)(2 110 63 14)(3 105 64 9)(4 108 57 12)(5 111 58 15)(6 106 59 10)(7 109 60 13)(8 112 61 16)(17 84 97 41)(18 87 98 44)(19 82 99 47)(20 85 100 42)(21 88 101 45)(22 83 102 48)(23 86 103 43)(24 81 104 46)(25 76 89 33)(26 79 90 36)(27 74 91 39)(28 77 92 34)(29 80 93 37)(30 75 94 40)(31 78 95 35)(32 73 96 38)(49 115 72 125)(50 118 65 128)(51 113 66 123)(52 116 67 126)(53 119 68 121)(54 114 69 124)(55 117 70 127)(56 120 71 122)
(1 74 123 99)(2 40 124 20)(3 76 125 101)(4 34 126 22)(5 78 127 103)(6 36 128 24)(7 80 121 97)(8 38 122 18)(9 25 72 88)(10 90 65 46)(11 27 66 82)(12 92 67 48)(13 29 68 84)(14 94 69 42)(15 31 70 86)(16 96 71 44)(17 60 37 119)(19 62 39 113)(21 64 33 115)(23 58 35 117)(26 50 81 106)(28 52 83 108)(30 54 85 110)(32 56 87 112)(41 109 93 53)(43 111 95 55)(45 105 89 49)(47 107 91 51)(57 77 116 102)(59 79 118 104)(61 73 120 98)(63 75 114 100)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim11111112222244
type++++++-++-
imageC1C2C2C2C2C2C4Q8D4SD16C4○D4C4○D8C8⋊C22C8.C22
kernelC4.Q810C4C22.7C42C22.4Q16C4×C4⋊C4C23.65C23C2×C4.Q8C4.Q8C4⋊C4C22×C4C2×C4C2×C4C22C22C22
# reps11311182248411

Matrix representation of C4.Q810C4 in GL6(𝔽17)

100000
010000
0016000
0001600
0000115
0000116
,
10160000
1670000
0001600
0016000
0000010
00001210
,
010000
1600000
0013000
0001300
0000712
00001310
,
0130000
400000
000100
0016000
0000160
0000016

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

C4.Q810C4 in GAP, Magma, Sage, TeX

C_4.Q_8\rtimes_{10}C_4
% in TeX

G:=Group("C4.Q8:10C4");
// GroupNames label

G:=SmallGroup(128,652);
// by ID

G=gap.SmallGroup(128,652);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,58,2804,718,172]);
// Polycyclic

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

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