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G = C42.31Q8order 128 = 27

31st non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.31Q8, C4⋊C89C4, C4⋊C4.17Q8, C4.37(C4×Q8), C4⋊C4.219D4, C4.145(C4×D4), C4.18(C4⋊Q8), C2.6(D4.Q8), C2.6(Q8.Q8), C42.152(C2×C4), C23.794(C2×D4), (C22×C4).699D4, C428C4.10C2, C2.6(D4.2D4), C22.73(C4○D8), C2.6(Q8.D4), C4.17(C42.C2), C22.93(C8⋊C22), C22.4Q16.36C2, (C2×C42).312C22, (C22×C8).316C22, C22.81(C22⋊Q8), C22.140(C4⋊D4), (C22×C4).1398C23, C22.82(C8.C22), C2.15(M4(2)⋊C4), C2.12(C23.25D4), C2.16(C23.65C23), (C2×C4⋊C8).44C2, (C4×C4⋊C4).24C2, (C2×C4).55(C4⋊C4), (C2×C8).115(C2×C4), (C2×C4).210(C2×Q8), (C2×C4.Q8).21C2, (C2×C2.D8).10C2, (C2×C4).1015(C2×D4), (C2×C4⋊C4).80C22, C22.116(C2×C4⋊C4), (C2×C4).762(C4○D4), (C2×C4).554(C22×C4), SmallGroup(128,681)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.31Q8
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.31Q8
C1C2C2×C4 — C42.31Q8
C1C23C2×C42 — C42.31Q8
C1C2C2C22×C4 — C42.31Q8

Generators and relations for C42.31Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=b-1c-1 >

Subgroups: 220 in 120 conjugacy classes, 64 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], C23, C42 [×4], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42 [×3], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×4], C22×C8 [×2], C22.4Q16 [×2], C4×C4⋊C4, C428C4, C2×C4⋊C8, C2×C4.Q8, C2×C2.D8, C42.31Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C4○D8 [×2], C8⋊C22, C8.C22, C23.65C23, C23.25D4, M4(2)⋊C4, D4.2D4, Q8.D4, D4.Q8, Q8.Q8, C42.31Q8

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111122222244
type+++++++-+-++-
imageC1C2C2C2C2C2C2C4Q8D4Q8D4C4○D4C4○D8C8⋊C22C8.C22
kernelC42.31Q8C22.4Q16C4×C4⋊C4C428C4C2×C4⋊C8C2×C4.Q8C2×C2.D8C4⋊C8C42C4⋊C4C4⋊C4C22×C4C2×C4C22C22C22
# reps1211111822224811

Matrix representation of C42.31Q8 in GL6(𝔽17)

400000
0130000
001200
00161600
000040
000004
,
1600000
0160000
001000
000100
0000613
0000511
,
1600000
010000
001200
00161600
0000114
000046
,
010000
100000
00101000
0012700
000045
00001413

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,5,0,0,0,0,13,11],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,4,0,0,0,0,4,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,0,0,10,7,0,0,0,0,0,0,4,14,0,0,0,0,5,13] >;

C42.31Q8 in GAP, Magma, Sage, TeX

C_4^2._{31}Q_8
% in TeX

G:=Group("C4^2.31Q8");
// GroupNames label

G:=SmallGroup(128,681);
// by ID

G=gap.SmallGroup(128,681);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,436,2019,1018,248]);
// Polycyclic

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

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