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G = C428C8order 128 = 27

5th semidirect product of C42 and C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C428C8, C43.8C2, C42.54Q8, C42.321D4, C4.12(C4⋊C8), (C2×C42).44C4, (C2×C4).77M4(2), C2.3(C428C4), C4.80(C4.4D4), (C22×C8).19C22, C22.36(C22×C8), C4.34(C42.C2), C2.4(C4⋊M4(2)), C2.6(C42.6C4), C23.265(C22×C4), C22.47(C2×M4(2)), C2.7(C42.12C4), (C22×C4).1621C23, (C2×C42).1054C22, C22.7C42.3C2, C22.56(C42⋊C2), C2.9(C2×C4⋊C8), (C2×C4⋊C8).24C2, (C2×C4).61(C2×C8), C22.64(C2×C4⋊C4), (C2×C4).337(C2×Q8), (C2×C4).129(C4⋊C4), (C2×C4).1519(C2×D4), (C2×C4).929(C4○D4), (C22×C4).443(C2×C4), SmallGroup(128,563)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C428C8
C1C2C4C2×C4C22×C4C2×C42C43 — C428C8
C1C22 — C428C8
C1C22×C4 — C428C8
C1C2C2C22×C4 — C428C8

Generators and relations for C428C8
 G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab2, cbc-1=a2b >

Subgroups: 188 in 130 conjugacy classes, 80 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×20], C2×C4 [×10], C23, C42 [×8], C42 [×10], C2×C8 [×12], C22×C4 [×3], C22×C4 [×4], C4⋊C8 [×4], C2×C42 [×3], C2×C42 [×4], C22×C8 [×4], C22.7C42 [×4], C43, C2×C4⋊C8 [×2], C428C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×6], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C4⋊C8 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C22×C8, C2×M4(2) [×3], C428C4, C2×C4⋊C8, C4⋊M4(2), C42.12C4 [×2], C42.6C4 [×2], C428C8

Smallest permutation representation of C428C8
Regular action on 128 points
Generators in S128
(1 115 103 91)(2 60 104 36)(3 117 97 93)(4 62 98 38)(5 119 99 95)(6 64 100 40)(7 113 101 89)(8 58 102 34)(9 86 70 110)(10 31 71 55)(11 88 72 112)(12 25 65 49)(13 82 66 106)(14 27 67 51)(15 84 68 108)(16 29 69 53)(17 61 41 37)(18 118 42 94)(19 63 43 39)(20 120 44 96)(21 57 45 33)(22 114 46 90)(23 59 47 35)(24 116 48 92)(26 126 50 78)(28 128 52 80)(30 122 54 74)(32 124 56 76)(73 85 121 109)(75 87 123 111)(77 81 125 105)(79 83 127 107)
(1 27 23 83)(2 52 24 108)(3 29 17 85)(4 54 18 110)(5 31 19 87)(6 56 20 112)(7 25 21 81)(8 50 22 106)(9 62 74 118)(10 39 75 95)(11 64 76 120)(12 33 77 89)(13 58 78 114)(14 35 79 91)(15 60 80 116)(16 37 73 93)(26 46 82 102)(28 48 84 104)(30 42 86 98)(32 44 88 100)(34 126 90 66)(36 128 92 68)(38 122 94 70)(40 124 96 72)(41 109 97 53)(43 111 99 55)(45 105 101 49)(47 107 103 51)(57 125 113 65)(59 127 115 67)(61 121 117 69)(63 123 119 71)
(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)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4AF8A···8P
order12···24···44···48···8
size11···11···12···24···4

56 irreducible representations

dim1111112222
type+++++-
imageC1C2C2C2C4C8D4Q8M4(2)C4○D4
kernelC428C8C22.7C42C43C2×C4⋊C8C2×C42C42C42C42C2×C4C2×C4
# reps141281622128

Matrix representation of C428C8 in GL5(𝔽17)

10000
00100
016000
00040
00004
,
160000
001600
01000
000016
000160
,
90000
06600
061100
000155
000122

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

C428C8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_8C_8
% in TeX

G:=Group("C4^2:8C8");
// GroupNames label

G:=SmallGroup(128,563);
// by ID

G=gap.SmallGroup(128,563);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,124]);
// Polycyclic

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

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