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G = C42.99D4order 128 = 27

81st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.99D4, Q81(C4⋊C4), (C4×Q8)⋊13C4, (C2×Q8).16Q8, (C2×C4).60Q16, C43(Q8⋊C4), (C2×Q8).157D4, C42.137(C2×C4), C429C4.3C2, C2.1(C4.Q16), C2.1(C4⋊SD16), C2.1(Q8⋊Q8), C2.1(C42Q16), (C2×C4).119SD16, C23.741(C2×D4), (C22×C4).267D4, C4.94(C22⋊Q8), C22.24(C2×Q16), C4.119(C4⋊D4), C22.4Q16.1C2, (C22×C8).11C22, C22.47(C2×SD16), C4.32(C42⋊C2), C22.60(C8⋊C22), (C2×C42).248C22, C22.68(C22⋊Q8), C22.106(C4⋊D4), (C22×C4).1325C23, C22.49(C8.C22), C2.12(C23.7Q8), (C22×Q8).383C22, C2.22(C23.36D4), C4.2(C2×C4⋊C4), (C2×C4⋊C8).22C2, (C2×C4×Q8).11C2, C4⋊C4.190(C2×C4), (C2×C4).260(C2×Q8), (C2×C4).1315(C2×D4), (C2×C4⋊C4).33C22, (C2×Q8).188(C2×C4), (C2×Q8⋊C4).1C2, C2.18(C2×Q8⋊C4), (C2×C4).861(C4○D4), (C2×C4).363(C22×C4), (C2×C4).359(C22⋊C4), C22.248(C2×C22⋊C4), SmallGroup(128,535)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.99D4
C1C2C22C2×C4C22×C4C22×Q8C2×C4×Q8 — C42.99D4
C1C2C2×C4 — C42.99D4
C1C23C2×C42 — C42.99D4
C1C2C2C22×C4 — C42.99D4

Generators and relations for C42.99D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 276 in 154 conjugacy classes, 76 normal (36 characteristic)
C1, C2 [×7], C4 [×4], C4 [×4], C4 [×10], C22 [×7], C8 [×2], C2×C4 [×6], C2×C4 [×8], C2×C4 [×18], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×3], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C4×Q8 [×4], C4×Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×2], C429C4, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C4×Q8, C42.99D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C2×SD16, C2×Q16, C8⋊C22, C8.C22, C23.7Q8, C2×Q8⋊C4, C23.36D4, C4⋊SD16, C42Q16, Q8⋊Q8, C4.Q16, C42.99D4

Smallest permutation representation of C42.99D4
Regular action on 128 points
Generators in S128
(1 87 39 127)(2 128 40 88)(3 81 33 121)(4 122 34 82)(5 83 35 123)(6 124 36 84)(7 85 37 125)(8 126 38 86)(9 17 52 28)(10 29 53 18)(11 19 54 30)(12 31 55 20)(13 21 56 32)(14 25 49 22)(15 23 50 26)(16 27 51 24)(41 64 113 97)(42 98 114 57)(43 58 115 99)(44 100 116 59)(45 60 117 101)(46 102 118 61)(47 62 119 103)(48 104 120 63)(65 74 106 94)(66 95 107 75)(67 76 108 96)(68 89 109 77)(69 78 110 90)(70 91 111 79)(71 80 112 92)(72 93 105 73)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 31 77 98)(2 101 78 26)(3 29 79 104)(4 99 80 32)(5 27 73 102)(6 97 74 30)(7 25 75 100)(8 103 76 28)(9 86 47 67)(10 70 48 81)(11 84 41 65)(12 68 42 87)(13 82 43 71)(14 66 44 85)(15 88 45 69)(16 72 46 83)(17 38 62 96)(18 91 63 33)(19 36 64 94)(20 89 57 39)(21 34 58 92)(22 95 59 37)(23 40 60 90)(24 93 61 35)(49 107 116 125)(50 128 117 110)(51 105 118 123)(52 126 119 108)(53 111 120 121)(54 124 113 106)(55 109 114 127)(56 122 115 112)
(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)| (1,87,39,127)(2,128,40,88)(3,81,33,121)(4,122,34,82)(5,83,35,123)(6,124,36,84)(7,85,37,125)(8,126,38,86)(9,17,52,28)(10,29,53,18)(11,19,54,30)(12,31,55,20)(13,21,56,32)(14,25,49,22)(15,23,50,26)(16,27,51,24)(41,64,113,97)(42,98,114,57)(43,58,115,99)(44,100,116,59)(45,60,117,101)(46,102,118,61)(47,62,119,103)(48,104,120,63)(65,74,106,94)(66,95,107,75)(67,76,108,96)(68,89,109,77)(69,78,110,90)(70,91,111,79)(71,80,112,92)(72,93,105,73), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,31,77,98)(2,101,78,26)(3,29,79,104)(4,99,80,32)(5,27,73,102)(6,97,74,30)(7,25,75,100)(8,103,76,28)(9,86,47,67)(10,70,48,81)(11,84,41,65)(12,68,42,87)(13,82,43,71)(14,66,44,85)(15,88,45,69)(16,72,46,83)(17,38,62,96)(18,91,63,33)(19,36,64,94)(20,89,57,39)(21,34,58,92)(22,95,59,37)(23,40,60,90)(24,93,61,35)(49,107,116,125)(50,128,117,110)(51,105,118,123)(52,126,119,108)(53,111,120,121)(54,124,113,106)(55,109,114,127)(56,122,115,112), (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( (1,87,39,127)(2,128,40,88)(3,81,33,121)(4,122,34,82)(5,83,35,123)(6,124,36,84)(7,85,37,125)(8,126,38,86)(9,17,52,28)(10,29,53,18)(11,19,54,30)(12,31,55,20)(13,21,56,32)(14,25,49,22)(15,23,50,26)(16,27,51,24)(41,64,113,97)(42,98,114,57)(43,58,115,99)(44,100,116,59)(45,60,117,101)(46,102,118,61)(47,62,119,103)(48,104,120,63)(65,74,106,94)(66,95,107,75)(67,76,108,96)(68,89,109,77)(69,78,110,90)(70,91,111,79)(71,80,112,92)(72,93,105,73), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,31,77,98)(2,101,78,26)(3,29,79,104)(4,99,80,32)(5,27,73,102)(6,97,74,30)(7,25,75,100)(8,103,76,28)(9,86,47,67)(10,70,48,81)(11,84,41,65)(12,68,42,87)(13,82,43,71)(14,66,44,85)(15,88,45,69)(16,72,46,83)(17,38,62,96)(18,91,63,33)(19,36,64,94)(20,89,57,39)(21,34,58,92)(22,95,59,37)(23,40,60,90)(24,93,61,35)(49,107,116,125)(50,128,117,110)(51,105,118,123)(52,126,119,108)(53,111,120,121)(54,124,113,106)(55,109,114,127)(56,122,115,112), (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([(1,87,39,127),(2,128,40,88),(3,81,33,121),(4,122,34,82),(5,83,35,123),(6,124,36,84),(7,85,37,125),(8,126,38,86),(9,17,52,28),(10,29,53,18),(11,19,54,30),(12,31,55,20),(13,21,56,32),(14,25,49,22),(15,23,50,26),(16,27,51,24),(41,64,113,97),(42,98,114,57),(43,58,115,99),(44,100,116,59),(45,60,117,101),(46,102,118,61),(47,62,119,103),(48,104,120,63),(65,74,106,94),(66,95,107,75),(67,76,108,96),(68,89,109,77),(69,78,110,90),(70,91,111,79),(71,80,112,92),(72,93,105,73)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128)], [(1,31,77,98),(2,101,78,26),(3,29,79,104),(4,99,80,32),(5,27,73,102),(6,97,74,30),(7,25,75,100),(8,103,76,28),(9,86,47,67),(10,70,48,81),(11,84,41,65),(12,68,42,87),(13,82,43,71),(14,66,44,85),(15,88,45,69),(16,72,46,83),(17,38,62,96),(18,91,63,33),(19,36,64,94),(20,89,57,39),(21,34,58,92),(22,95,59,37),(23,40,60,90),(24,93,61,35),(49,107,116,125),(50,128,117,110),(51,105,118,123),(52,126,119,108),(53,111,120,121),(54,124,113,106),(55,109,114,127),(56,122,115,112)], [(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)])

38 conjugacy classes

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

38 irreducible representations

dim1111111222222244
type+++++++++--+-
imageC1C2C2C2C2C2C4D4D4D4Q8SD16Q16C4○D4C8⋊C22C8.C22
kernelC42.99D4C22.4Q16C429C4C2×Q8⋊C4C2×C4⋊C8C2×C4×Q8C4×Q8C42C22×C4C2×Q8C2×Q8C2×C4C2×C4C2×C4C22C22
# reps1212118222244411

Matrix representation of C42.99D4 in GL6(𝔽17)

010000
1600000
001000
000100
0000160
0000016
,
100000
010000
001000
000100
000001
0000160
,
100000
0160000
00101200
0010700
0000016
0000160
,
1600000
010000
0071500
0071000
0000125
00001212

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,10,10,0,0,0,0,12,7,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,15,10,0,0,0,0,0,0,12,12,0,0,0,0,5,12] >;

C42.99D4 in GAP, Magma, Sage, TeX

C_4^2._{99}D_4
% in TeX

G:=Group("C4^2.99D4");
// GroupNames label

G:=SmallGroup(128,535);
// by ID

G=gap.SmallGroup(128,535);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,352,2804,718,172]);
// Polycyclic

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

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