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G = C42.8Q8order 128 = 27

8th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.8Q8, C42.29D4, C4⋊C87C4, (C2×C4).116D8, (C2×C4).8C42, (C2×C4).50Q16, C4.1(C2.D8), C4.1(C4.Q8), C42.36(C2×C4), (C2×C4).90SD16, C429C4.1C2, C2.2(C4.D8), (C22×C4).722D4, C4.30(D4⋊C4), C4.22(Q8⋊C4), C2.2(C4.10D8), C2.2(C4.6Q16), C2.7(C22.4Q16), (C2×C42).133C22, C22.39(D4⋊C4), C2.7(C22.C42), C23.216(C22⋊C4), C22.29(Q8⋊C4), C22.25(C4.D4), C22.16(C4.10D4), C22.46(C2.C42), (C2×C4⋊C4).5C4, (C2×C4⋊C8).6C2, (C2×C4).96(C4⋊C4), (C22×C4).151(C2×C4), (C2×C4).339(C22⋊C4), SmallGroup(128,28)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.8Q8
C1C2C22C2×C4C22×C4C2×C42C2×C4⋊C8 — C42.8Q8
C1C22C2×C4 — C42.8Q8
C1C23C2×C42 — C42.8Q8
C1C22C22C2×C42 — C42.8Q8

Generators and relations for C42.8Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a-1b2c3 >

Subgroups: 184 in 94 conjugacy classes, 56 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×4], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×8], C23, C42 [×2], C42 [×2], C4⋊C4 [×6], C2×C8 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C429C4, C2×C4⋊C8 [×2], C42.8Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8 [×2], SD16 [×4], Q16 [×2], C2.C42, C4.D4, C4.10D4, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C4.D8, C4.10D8 [×2], C4.6Q16, C22.4Q16 [×2], C22.C42, C42.8Q8

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

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I4J4K4L4M4N8A···8P
order12···24···44444448···8
size11···12···24488884···4

38 irreducible representations

dim1111122222244
type++++-++-+-
imageC1C2C2C4C4D4Q8D4D8SD16Q16C4.D4C4.10D4
kernelC42.8Q8C429C4C2×C4⋊C8C4⋊C8C2×C4⋊C4C42C42C22×C4C2×C4C2×C4C2×C4C22C22
# reps1128411248411

Matrix representation of C42.8Q8 in GL5(𝔽17)

160000
016000
001600
00012
0001616
,
10000
001600
01000
000160
000016
,
130000
012500
0121200
00082
000109
,
40000
01700
071600
000155
000132

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,8,10,0,0,0,2,9],[4,0,0,0,0,0,1,7,0,0,0,7,16,0,0,0,0,0,15,13,0,0,0,5,2] >;

C42.8Q8 in GAP, Magma, Sage, TeX

C_4^2._8Q_8
% in TeX

G:=Group("C4^2.8Q8");
// GroupNames label

G:=SmallGroup(128,28);
// by ID

G=gap.SmallGroup(128,28);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,3924,242]);
// 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=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^-1*b^2*c^3>;
// generators/relations

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