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G = C42.55Q8order 128 = 27

15th non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Aliases: C42.55Q8, (C4×C8)⋊14C4, (C2×C4).73D8, (C2×C4).35Q16, (C2×C4).70SD16, C4.11(C4.Q8), C4.15(C2.D8), C22.34(C2×D8), C429C4.4C2, C42.319(C2×C4), C2.1(C4.4D8), (C22×C4).576D4, C23.750(C2×D4), C4.1(C42.C2), C22.27(C2×Q16), C2.7(C428C4), C22.4Q16.3C2, C2.1(C4.SD16), C22.51(C2×SD16), C4.57(C42⋊C2), (C22×C8).478C22, (C22×C4).1338C23, (C2×C42).1055C22, C22.55(C4.4D4), (C2×C4×C8).17C2, C2.7(C2×C2.D8), C2.7(C2×C4.Q8), (C2×C8).211(C2×C4), C22.96(C2×C4⋊C4), (C2×C4).190(C2×Q8), (C2×C4).130(C4⋊C4), (C2×C4⋊C4).44C22, (C2×C4).556(C4○D4), (C2×C4).536(C22×C4), SmallGroup(128,566)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.55Q8
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — C42.55Q8
C1C2C2×C4 — C42.55Q8
C1C23C2×C42 — C42.55Q8
C1C2C2C22×C4 — C42.55Q8

Generators and relations for C42.55Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c3 >

Subgroups: 236 in 124 conjugacy classes, 76 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×4], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×12], C23, C42 [×4], C4⋊C4 [×16], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C4×C8 [×4], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C22×C8 [×2], C22.4Q16 [×4], C429C4 [×2], C2×C4×C8, C42.55Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C428C4, C2×C4.Q8, C2×C2.D8, C4.4D8 [×2], C4.SD16 [×2], C42.55Q8

Smallest permutation representation of C42.55Q8
Regular action on 128 points
Generators in S128
(1 75 17 62)(2 76 18 63)(3 77 19 64)(4 78 20 57)(5 79 21 58)(6 80 22 59)(7 73 23 60)(8 74 24 61)(9 50 42 32)(10 51 43 25)(11 52 44 26)(12 53 45 27)(13 54 46 28)(14 55 47 29)(15 56 48 30)(16 49 41 31)(33 91 121 71)(34 92 122 72)(35 93 123 65)(36 94 124 66)(37 95 125 67)(38 96 126 68)(39 89 127 69)(40 90 128 70)(81 111 119 101)(82 112 120 102)(83 105 113 103)(84 106 114 104)(85 107 115 97)(86 108 116 98)(87 109 117 99)(88 110 118 100)
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 59 13 63)(10 60 14 64)(11 61 15 57)(12 62 16 58)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)(33 104 37 100)(34 97 38 101)(35 98 39 102)(36 99 40 103)(41 79 45 75)(42 80 46 76)(43 73 47 77)(44 74 48 78)(65 116 69 120)(66 117 70 113)(67 118 71 114)(68 119 72 115)(81 92 85 96)(82 93 86 89)(83 94 87 90)(84 95 88 91)(105 124 109 128)(106 125 110 121)(107 126 111 122)(108 127 112 123)
(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 83 55 92)(2 120 56 71)(3 81 49 90)(4 118 50 69)(5 87 51 96)(6 116 52 67)(7 85 53 94)(8 114 54 65)(9 39 57 100)(10 126 58 109)(11 37 59 98)(12 124 60 107)(13 35 61 104)(14 122 62 105)(15 33 63 102)(16 128 64 111)(17 113 29 72)(18 82 30 91)(19 119 31 70)(20 88 32 89)(21 117 25 68)(22 86 26 95)(23 115 27 66)(24 84 28 93)(34 75 103 47)(36 73 97 45)(38 79 99 43)(40 77 101 41)(42 127 78 110)(44 125 80 108)(46 123 74 106)(48 121 76 112)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim11111222222
type++++-++-
imageC1C2C2C2C4Q8D4D8SD16Q16C4○D4
kernelC42.55Q8C22.4Q16C429C4C2×C4×C8C4×C8C42C22×C4C2×C4C2×C4C2×C4C2×C4
# reps14218224848

Matrix representation of C42.55Q8 in GL5(𝔽17)

160000
016000
001600
000013
000130
,
160000
0161500
01100
00010
00001
,
160000
00600
0141100
00040
00004
,
40000
015500
013200
000111
000616

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

C42.55Q8 in GAP, Magma, Sage, TeX

C_4^2._{55}Q_8
% in TeX

G:=Group("C4^2.55Q8");
// GroupNames label

G:=SmallGroup(128,566);
// by ID

G=gap.SmallGroup(128,566);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations

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