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G = C42.35Q8order 128 = 27

35th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.35Q8, C23.452C24, C22.2372+ 1+4, C22.1832- 1+4, C4⋊C4.23Q8, C4⋊C4.235D4, C4.27(C4⋊Q8), C43(C42.C2), C2.52(D46D4), C429C4.29C2, C2.25(Q86D4), C2.32(D43Q8), C2.17(Q83Q8), (C22×C4).97C23, (C2×C42).557C22, C22.303(C22×D4), C22.102(C22×Q8), C2.C42.189C22, C23.81C23.15C2, C23.65C23.54C2, C2.13(C23.41C23), (C4×C4⋊C4).65C2, C2.15(C2×C4⋊Q8), (C2×C4).78(C2×D4), (C2×C4).52(C2×Q8), C2.15(C2×C42.C2), (C2×C4).823(C4○D4), (C2×C4⋊C4).305C22, C22.328(C2×C4○D4), (C2×C42.C2).20C2, SmallGroup(128,1284)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.35Q8
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.35Q8
C1C23 — C42.35Q8
C1C23 — C42.35Q8
C1C23 — C42.35Q8

Generators and relations for C42.35Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 356 in 226 conjugacy classes, 124 normal (28 characteristic)
C1, C2 [×7], C4 [×8], C4 [×18], C22 [×7], C2×C4 [×22], C2×C4 [×34], C23, C42 [×4], C42 [×8], C4⋊C4 [×8], C4⋊C4 [×34], C22×C4 [×3], C22×C4 [×12], C2.C42 [×8], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×18], C42.C2 [×8], C4×C4⋊C4 [×2], C429C4, C429C4 [×2], C23.65C23 [×4], C23.81C23 [×4], C2×C42.C2 [×2], C42.35Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C42.C2 [×4], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ 1+4, 2- 1+4, C2×C42.C2, C2×C4⋊Q8, C23.41C23, D46D4, Q86D4, D43Q8, Q83Q8, C42.35Q8

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

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4Z4AA4AB4AC4AD
order12···24···44···44444
size11···12···24···48888

38 irreducible representations

dim111111222244
type++++++-+-+-
imageC1C2C2C2C2C2Q8D4Q8C4○D42+ 1+42- 1+4
kernelC42.35Q8C4×C4⋊C4C429C4C23.65C23C23.81C23C2×C42.C2C42C4⋊C4C4⋊C4C2×C4C22C22
# reps123442444811

Matrix representation of C42.35Q8 in GL6(𝔽5)

010000
400000
002000
000300
000003
000030
,
400000
040000
002000
000300
000010
000001
,
010000
100000
004000
000100
000030
000002
,
400000
040000
000100
004000
000002
000020

G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0] >;

C42.35Q8 in GAP, Magma, Sage, TeX

C_4^2._{35}Q_8
% in TeX

G:=Group("C4^2.35Q8");
// GroupNames label

G:=SmallGroup(128,1284);
// by ID

G=gap.SmallGroup(128,1284);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,268,675,80]);
// Polycyclic

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

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