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G = C424C8order 128 = 27

1st semidirect product of C42 and C8 acting via C8/C4=C2

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

Aliases: C424C8, C43.2C2, (C4×C8)⋊11C4, C4.14(C4×C8), (C2×C4).89C42, (C2×C42).30C4, C4.17(C8⋊C4), C42.339(C2×C4), (C2×C4).89M4(2), C2.2(C424C4), C22.18(C22×C8), C22.26(C2×C42), C4.67(C42⋊C2), (C2×C42).984C22, (C22×C8).468C22, C23.247(C22×C4), C22.34(C2×M4(2)), C2.1(C42.12C4), C42(C22.7C42), (C22×C4).1600C23, C22.45(C42⋊C2), C22.7C42.47C2, C42(C22.7C42), C2.5(C2×C4×C8), (C2×C4×C8).11C2, C2.4(C2×C8⋊C4), (C2×C4).58(C2×C8), (C2×C8).202(C2×C4), (C2×C4).910(C4○D4), (C22×C4).504(C2×C4), (C2×C4).590(C22×C4), (C2×C42)(C22.7C42), SmallGroup(128,476)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C424C8
C1C2C22C2×C4C22×C4C2×C42C43 — C424C8
C1C2 — C424C8
C1C2×C42 — C424C8
C1C2C2C22×C4 — C424C8

Generators and relations for C424C8
 G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab2, bc=cb >

Subgroups: 188 in 148 conjugacy classes, 108 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×26], C2×C4 [×8], C23, C42 [×12], C42 [×8], C2×C8 [×8], C2×C8 [×8], C22×C4 [×3], C22×C4 [×4], C4×C8 [×8], C2×C42, C2×C42 [×6], C22×C8 [×4], C22.7C42 [×4], C43, C2×C4×C8 [×2], C424C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], M4(2) [×4], C22×C4 [×3], C4○D4 [×4], C4×C8 [×4], C8⋊C4 [×4], C2×C42, C42⋊C2 [×6], C22×C8 [×2], C2×M4(2) [×2], C424C4, C2×C4×C8, C2×C8⋊C4, C42.12C4 [×4], C424C8

Smallest permutation representation of C424C8
Regular action on 128 points
Generators in S128
(1 117 99 91)(2 62 100 36)(3 119 101 93)(4 64 102 38)(5 113 103 95)(6 58 104 40)(7 115 97 89)(8 60 98 34)(9 32 124 50)(10 81 125 107)(11 26 126 52)(12 83 127 109)(13 28 128 54)(14 85 121 111)(15 30 122 56)(16 87 123 105)(17 61 43 35)(18 118 44 92)(19 63 45 37)(20 120 46 94)(21 57 47 39)(22 114 48 96)(23 59 41 33)(24 116 42 90)(25 67 51 79)(27 69 53 73)(29 71 55 75)(31 65 49 77)(66 106 78 88)(68 108 80 82)(70 110 74 84)(72 112 76 86)
(1 29 17 85)(2 30 18 86)(3 31 19 87)(4 32 20 88)(5 25 21 81)(6 26 22 82)(7 27 23 83)(8 28 24 84)(9 94 78 38)(10 95 79 39)(11 96 80 40)(12 89 73 33)(13 90 74 34)(14 91 75 35)(15 92 76 36)(16 93 77 37)(41 109 97 53)(42 110 98 54)(43 111 99 55)(44 112 100 56)(45 105 101 49)(46 106 102 50)(47 107 103 51)(48 108 104 52)(57 125 113 67)(58 126 114 68)(59 127 115 69)(60 128 116 70)(61 121 117 71)(62 122 118 72)(63 123 119 65)(64 124 120 66)
(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)| (1,117,99,91)(2,62,100,36)(3,119,101,93)(4,64,102,38)(5,113,103,95)(6,58,104,40)(7,115,97,89)(8,60,98,34)(9,32,124,50)(10,81,125,107)(11,26,126,52)(12,83,127,109)(13,28,128,54)(14,85,121,111)(15,30,122,56)(16,87,123,105)(17,61,43,35)(18,118,44,92)(19,63,45,37)(20,120,46,94)(21,57,47,39)(22,114,48,96)(23,59,41,33)(24,116,42,90)(25,67,51,79)(27,69,53,73)(29,71,55,75)(31,65,49,77)(66,106,78,88)(68,108,80,82)(70,110,74,84)(72,112,76,86), (1,29,17,85)(2,30,18,86)(3,31,19,87)(4,32,20,88)(5,25,21,81)(6,26,22,82)(7,27,23,83)(8,28,24,84)(9,94,78,38)(10,95,79,39)(11,96,80,40)(12,89,73,33)(13,90,74,34)(14,91,75,35)(15,92,76,36)(16,93,77,37)(41,109,97,53)(42,110,98,54)(43,111,99,55)(44,112,100,56)(45,105,101,49)(46,106,102,50)(47,107,103,51)(48,108,104,52)(57,125,113,67)(58,126,114,68)(59,127,115,69)(60,128,116,70)(61,121,117,71)(62,122,118,72)(63,123,119,65)(64,124,120,66), (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( (1,117,99,91)(2,62,100,36)(3,119,101,93)(4,64,102,38)(5,113,103,95)(6,58,104,40)(7,115,97,89)(8,60,98,34)(9,32,124,50)(10,81,125,107)(11,26,126,52)(12,83,127,109)(13,28,128,54)(14,85,121,111)(15,30,122,56)(16,87,123,105)(17,61,43,35)(18,118,44,92)(19,63,45,37)(20,120,46,94)(21,57,47,39)(22,114,48,96)(23,59,41,33)(24,116,42,90)(25,67,51,79)(27,69,53,73)(29,71,55,75)(31,65,49,77)(66,106,78,88)(68,108,80,82)(70,110,74,84)(72,112,76,86), (1,29,17,85)(2,30,18,86)(3,31,19,87)(4,32,20,88)(5,25,21,81)(6,26,22,82)(7,27,23,83)(8,28,24,84)(9,94,78,38)(10,95,79,39)(11,96,80,40)(12,89,73,33)(13,90,74,34)(14,91,75,35)(15,92,76,36)(16,93,77,37)(41,109,97,53)(42,110,98,54)(43,111,99,55)(44,112,100,56)(45,105,101,49)(46,106,102,50)(47,107,103,51)(48,108,104,52)(57,125,113,67)(58,126,114,68)(59,127,115,69)(60,128,116,70)(61,121,117,71)(62,122,118,72)(63,123,119,65)(64,124,120,66), (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([(1,117,99,91),(2,62,100,36),(3,119,101,93),(4,64,102,38),(5,113,103,95),(6,58,104,40),(7,115,97,89),(8,60,98,34),(9,32,124,50),(10,81,125,107),(11,26,126,52),(12,83,127,109),(13,28,128,54),(14,85,121,111),(15,30,122,56),(16,87,123,105),(17,61,43,35),(18,118,44,92),(19,63,45,37),(20,120,46,94),(21,57,47,39),(22,114,48,96),(23,59,41,33),(24,116,42,90),(25,67,51,79),(27,69,53,73),(29,71,55,75),(31,65,49,77),(66,106,78,88),(68,108,80,82),(70,110,74,84),(72,112,76,86)], [(1,29,17,85),(2,30,18,86),(3,31,19,87),(4,32,20,88),(5,25,21,81),(6,26,22,82),(7,27,23,83),(8,28,24,84),(9,94,78,38),(10,95,79,39),(11,96,80,40),(12,89,73,33),(13,90,74,34),(14,91,75,35),(15,92,76,36),(16,93,77,37),(41,109,97,53),(42,110,98,54),(43,111,99,55),(44,112,100,56),(45,105,101,49),(46,106,102,50),(47,107,103,51),(48,108,104,52),(57,125,113,67),(58,126,114,68),(59,127,115,69),(60,128,116,70),(61,121,117,71),(62,122,118,72),(63,123,119,65),(64,124,120,66)], [(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)])

80 conjugacy classes

class 1 2A···2G4A···4X4Y···4AN8A···8AF
order12···24···44···48···8
size11···11···12···22···2

80 irreducible representations

dim111111122
type++++
imageC1C2C2C2C4C4C8M4(2)C4○D4
kernelC424C8C22.7C42C43C2×C4×C8C4×C8C2×C42C42C2×C4C2×C4
# reps14121683288

Matrix representation of C424C8 in GL4(𝔽17) generated by

16000
01300
00413
00013
,
16000
0100
0040
0004
,
15000
01300
00161
00151
G:=sub<GL(4,GF(17))| [16,0,0,0,0,13,0,0,0,0,4,0,0,0,13,13],[16,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[15,0,0,0,0,13,0,0,0,0,16,15,0,0,1,1] >;

C424C8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4C_8
% in TeX

G:=Group("C4^2:4C8");
// GroupNames label

G:=SmallGroup(128,476);
// by ID

G=gap.SmallGroup(128,476);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,58,172]);
// Polycyclic

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

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