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G = C8⋊(C4⋊C4)  order 128 = 27

3rd semidirect product of C8 and C4⋊C4 acting via C4⋊C4/C22=C22

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

Aliases: C83(C4⋊C4), C4.Q87C4, C4.9(C4×Q8), C2.3(C8⋊Q8), (C2×C8).13Q8, (C2×C8).111D4, C2.4(C82D4), C2.4(C8.D4), C23.791(C2×D4), C22.177(C4×D4), (C22×C4).133D4, C4.77(C22⋊Q8), C22.34(C4⋊Q8), C4.7(C42.C2), C22.90(C8⋊C22), C22.4Q16.50C2, (C2×C42).308C22, (C22×C8).401C22, C2.18(SD16⋊C4), C22.137(C4⋊D4), (C22×C4).1393C23, C22.79(C8.C22), C23.65C23.13C2, C2.11(C23.65C23), C4.42(C2×C4⋊C4), C4⋊C4.90(C2×C4), (C2×C8).66(C2×C4), (C2×C8⋊C4).7C2, (C2×C4.Q8).6C2, (C2×C4).205(C2×Q8), (C2×C2.D8).35C2, (C2×C4).1350(C2×D4), (C2×C4⋊C4).77C22, (C2×C4).588(C4○D4), (C2×C4).411(C22×C4), SmallGroup(128,676)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8⋊(C4⋊C4)
C1C2C22C2×C4C22×C4C22×C8C2×C8⋊C4 — C8⋊(C4⋊C4)
C1C2C2×C4 — C8⋊(C4⋊C4)
C1C23C2×C42 — C8⋊(C4⋊C4)
C1C2C2C22×C4 — C8⋊(C4⋊C4)

Generators and relations for C8⋊(C4⋊C4)
 G = < a,b,c | a8=b4=c4=1, bab-1=a3, cac-1=a5, cbc-1=b-1 >

Subgroups: 228 in 120 conjugacy classes, 64 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×20], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×2], C4.Q8 [×4], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×2], C23.65C23 [×2], C2×C8⋊C4, C2×C4.Q8, C2×C2.D8, C8⋊(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C8⋊C22 [×2], C8.C22 [×2], C23.65C23, SD16⋊C4 [×2], C82D4, C8.D4, C8⋊Q8 [×2], C8⋊(C4⋊C4)

Smallest permutation representation of C8⋊(C4⋊C4)
Regular action on 128 points
Generators in S128
(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)
(1 103 19 118)(2 98 20 113)(3 101 21 116)(4 104 22 119)(5 99 23 114)(6 102 24 117)(7 97 17 120)(8 100 18 115)(9 27 43 124)(10 30 44 127)(11 25 45 122)(12 28 46 125)(13 31 47 128)(14 26 48 123)(15 29 41 126)(16 32 42 121)(33 84 52 70)(34 87 53 65)(35 82 54 68)(36 85 55 71)(37 88 56 66)(38 83 49 69)(39 86 50 72)(40 81 51 67)(57 108 75 91)(58 111 76 94)(59 106 77 89)(60 109 78 92)(61 112 79 95)(62 107 80 90)(63 110 73 93)(64 105 74 96)
(1 50 42 73)(2 55 43 78)(3 52 44 75)(4 49 45 80)(5 54 46 77)(6 51 47 74)(7 56 48 79)(8 53 41 76)(9 60 20 36)(10 57 21 33)(11 62 22 38)(12 59 23 35)(13 64 24 40)(14 61 17 37)(15 58 18 34)(16 63 19 39)(25 90 119 69)(26 95 120 66)(27 92 113 71)(28 89 114 68)(29 94 115 65)(30 91 116 70)(31 96 117 67)(32 93 118 72)(81 128 105 102)(82 125 106 99)(83 122 107 104)(84 127 108 101)(85 124 109 98)(86 121 110 103)(87 126 111 100)(88 123 112 97)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111222244
type+++++++-++-
imageC1C2C2C2C2C2C4D4Q8D4C4○D4C8⋊C22C8.C22
kernelC8⋊(C4⋊C4)C22.4Q16C23.65C23C2×C8⋊C4C2×C4.Q8C2×C2.D8C4.Q8C2×C8C2×C8C22×C4C2×C4C22C22
# reps1221118242422

Matrix representation of C8⋊(C4⋊C4) in GL8(𝔽17)

016000000
10000000
00100000
00010000
000000160
000000016
00000100
000016000
,
33000000
314000000
00400000
0013130000
00007848
0000810813
0000139810
000094109
,
130000000
013000000
0015130000
00520000
0000136162
000011131516
00001516411
000011564

G:=sub<GL(8,GF(17))| [0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[3,3,0,0,0,0,0,0,3,14,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,7,8,13,9,0,0,0,0,8,10,9,4,0,0,0,0,4,8,8,10,0,0,0,0,8,13,10,9],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,15,5,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,13,11,15,1,0,0,0,0,6,13,16,15,0,0,0,0,16,15,4,6,0,0,0,0,2,16,11,4] >;

C8⋊(C4⋊C4) in GAP, Magma, Sage, TeX

C_8\rtimes (C_4\rtimes C_4)
% in TeX

G:=Group("C8:(C4:C4)");
// GroupNames label

G:=SmallGroup(128,676);
// by ID

G=gap.SmallGroup(128,676);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,436,2804,172]);
// Polycyclic

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

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