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G = C87(C4⋊C4)  order 128 = 27

4th semidirect product of C8 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

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

Aliases: C87(C4⋊C4), C4.6(C4×Q8), C4.Q812C4, (C2×C8).54Q8, (C2×C8).368D4, C2.6(C88D4), C2.2(C83Q8), (C2×C4).63SD16, C2.18(C4×SD16), (C22×C4).555D4, C22.174(C4×D4), C23.788(C2×D4), C4.74(C22⋊Q8), C22.31(C4⋊Q8), C2.2(C8.5Q8), C4.4(C42.C2), C22.71(C4○D8), C22.68(C2×SD16), C22.4Q16.14C2, (C22×C8).486C22, C22.134(C4⋊D4), (C22×C4).1390C23, (C2×C42).1067C22, C23.65C23.10C2, C2.8(C23.65C23), (C2×C4×C8).55C2, C4.39(C2×C4⋊C4), C4⋊C4.87(C2×C4), (C2×C8).183(C2×C4), (C2×C4).202(C2×Q8), (C2×C4.Q8).19C2, (C2×C4).1347(C2×D4), (C2×C4⋊C4).74C22, (C2×C4).585(C4○D4), (C2×C4).408(C22×C4), SmallGroup(128,673)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C87(C4⋊C4)
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — C87(C4⋊C4)
C1C2C2×C4 — C87(C4⋊C4)
C1C23C2×C42 — C87(C4⋊C4)
C1C2C2C22×C4 — C87(C4⋊C4)

Generators and relations for C87(C4⋊C4)
 G = < a,b,c | a8=b4=c4=1, bab-1=a3, ac=ca, cbc-1=b-1 >

Subgroups: 228 in 124 conjugacy classes, 68 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C4×C8 [×2], C4.Q8 [×4], C4.Q8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×2], C23.65C23 [×2], C2×C4×C8, C2×C4.Q8 [×2], C87(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], SD16 [×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, C2×SD16 [×2], C4○D8 [×2], C23.65C23, C4×SD16 [×2], C88D4 [×2], C83Q8, C8.5Q8, C87(C4⋊C4)

Smallest permutation representation of C87(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)
(1 100 23 114)(2 103 24 117)(3 98 17 120)(4 101 18 115)(5 104 19 118)(6 99 20 113)(7 102 21 116)(8 97 22 119)(9 28 46 124)(10 31 47 127)(11 26 48 122)(12 29 41 125)(13 32 42 128)(14 27 43 123)(15 30 44 126)(16 25 45 121)(33 81 51 67)(34 84 52 70)(35 87 53 65)(36 82 54 68)(37 85 55 71)(38 88 56 66)(39 83 49 69)(40 86 50 72)(57 109 75 91)(58 112 76 94)(59 107 77 89)(60 110 78 92)(61 105 79 95)(62 108 80 90)(63 111 73 93)(64 106 74 96)
(1 54 41 73)(2 55 42 74)(3 56 43 75)(4 49 44 76)(5 50 45 77)(6 51 46 78)(7 52 47 79)(8 53 48 80)(9 60 20 33)(10 61 21 34)(11 62 22 35)(12 63 23 36)(13 64 24 37)(14 57 17 38)(15 58 18 39)(16 59 19 40)(25 89 118 72)(26 90 119 65)(27 91 120 66)(28 92 113 67)(29 93 114 68)(30 94 115 69)(31 95 116 70)(32 96 117 71)(81 124 110 99)(82 125 111 100)(83 126 112 101)(84 127 105 102)(85 128 106 103)(86 121 107 104)(87 122 108 97)(88 123 109 98)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4T8A···8P
order12···24···44···48···8
size11···12···28···82···2

44 irreducible representations

dim111111222222
type++++++-+
imageC1C2C2C2C2C4D4Q8D4SD16C4○D4C4○D8
kernelC87(C4⋊C4)C22.4Q16C23.65C23C2×C4×C8C2×C4.Q8C4.Q8C2×C8C2×C8C22×C4C2×C4C2×C4C22
# reps122128242848

Matrix representation of C87(C4⋊C4) in GL5(𝔽17)

10000
015000
00900
000150
00009
,
160000
001600
01000
00004
000130
,
130000
016000
00100
000130
000013

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

C87(C4⋊C4) in GAP, Magma, Sage, TeX

C_8\rtimes_7(C_4\rtimes C_4)
% in TeX

G:=Group("C8:7(C4:C4)");
// GroupNames label

G:=SmallGroup(128,673);
// by ID

G=gap.SmallGroup(128,673);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2804,172]);
// Polycyclic

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

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