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G = C42.24Q8order 128 = 27

24th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.24Q8, C8⋊C416C4, C429C4.5C2, C42.142(C2×C4), C428C4.7C2, (C22×C4).279D4, C23.752(C2×D4), C4.3(C42.C2), C2.9(C428C4), C4.59(C42⋊C2), C22.65(C8⋊C22), C22.4Q16.47C2, (C2×C42).255C22, (C22×C8).385C22, C2.8(M4(2)⋊C4), (C22×C4).1340C23, C22.57(C4.4D4), C22.54(C8.C22), C2.1(C42.28C22), C2.1(C42.30C22), C2.1(C42.29C22), (C2×C4).45(C4⋊C4), (C2×C8).144(C2×C4), C22.98(C2×C4⋊C4), (C2×C4).192(C2×Q8), (C2×C8⋊C4).28C2, (C2×C4⋊C4).46C22, (C2×C4).558(C4○D4), (C2×C4).538(C22×C4), SmallGroup(128,568)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.24Q8
C1C2C22C2×C4C22×C4C22×C8C2×C8⋊C4 — C42.24Q8
C1C2C2×C4 — C42.24Q8
C1C23C2×C42 — C42.24Q8
C1C2C2C22×C4 — C42.24Q8

Generators and relations for C42.24Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2c3 >

Subgroups: 220 in 112 conjugacy classes, 60 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C4⋊C4 [×12], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×4], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×4], C428C4, C429C4, C2×C8⋊C4, C42.24Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C8⋊C22 [×2], C8.C22 [×2], C428C4, M4(2)⋊C4 [×2], C42.28C22 [×2], C42.29C22, C42.30C22, C42.24Q8

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

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim11111122244
type+++++-++-
imageC1C2C2C2C2C4Q8D4C4○D4C8⋊C22C8.C22
kernelC42.24Q8C22.4Q16C428C4C429C4C2×C8⋊C4C8⋊C4C42C22×C4C2×C4C22C22
# reps14111822822

Matrix representation of C42.24Q8 in GL8(𝔽17)

19000000
1316000000
00190000
0013160000
00000200
00009000
000000015
00000080
,
10000000
01000000
001600000
000160000
00004000
00000400
000000130
000000013
,
42000000
113000000
00420000
001130000
00000100
00004000
00000004
000000160
,
103000000
17000000
00890000
00690000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(17))| [1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,15,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0],[10,1,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,8,6,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.24Q8 in GAP, Magma, Sage, TeX

C_4^2._{24}Q_8
% in TeX

G:=Group("C4^2.24Q8");
// GroupNames label

G:=SmallGroup(128,568);
// by ID

G=gap.SmallGroup(128,568);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,723,58,2019,248,2804,172]);
// 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,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^3>;
// generators/relations

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