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G = C42.36Q8order 128 = 27

36th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.36Q8, C23.470C24, C22.1912- 1+4, C22.2532+ 1+4, C429C4.32C2, C428C4.35C2, (C2×C42).67C22, C4.32(C42.C2), (C22×C4).103C23, C22.111(C22×Q8), C23.83C23.15C2, C23.65C23.56C2, C23.81C23.17C2, C23.63C23.28C2, C2.C42.206C22, C2.29(C23.37C23), C2.41(C22.50C24), C2.60(C22.47C24), C2.14(C23.41C23), C2.66(C22.46C24), C2.19(C22.53C24), (C4×C4⋊C4).69C2, (C2×C4).126(C2×Q8), C2.17(C2×C42.C2), (C2×C4).152(C4○D4), (C2×C4⋊C4).317C22, C22.346(C2×C4○D4), SmallGroup(128,1302)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.36Q8
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.36Q8
C1C23 — C42.36Q8
C1C23 — C42.36Q8
C1C23 — C42.36Q8

Generators and relations for C42.36Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 308 in 190 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C4 [×4], C4 [×18], C22 [×7], C2×C4 [×14], C2×C4 [×38], C23, C42 [×4], C42 [×6], C4⋊C4 [×24], C22×C4 [×7], C22×C4 [×8], C2.C42 [×2], C2.C42 [×14], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×6], C2×C4⋊C4 [×8], C4×C4⋊C4 [×2], C428C4 [×2], C429C4, C23.63C23 [×4], C23.65C23 [×2], C23.81C23 [×2], C23.83C23 [×2], C42.36Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C2×C42.C2, C23.37C23, C23.41C23, C22.46C24, C22.47C24, C22.50C24, C22.53C24, C42.36Q8

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111112244
type++++++++-+-
imageC1C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC42.36Q8C4×C4⋊C4C428C4C429C4C23.63C23C23.65C23C23.81C23C23.83C23C42C2×C4C22C22
# reps1221422241611

Matrix representation of C42.36Q8 in GL6(𝔽5)

210000
230000
004000
000400
000004
000010
,
420000
410000
001000
000100
000040
000004
,
130000
140000
004300
001100
000030
000002
,
210000
030000
002000
003300
000002
000030

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

C42.36Q8 in GAP, Magma, Sage, TeX

C_4^2._{36}Q_8
% in TeX

G:=Group("C4^2.36Q8");
// GroupNames label

G:=SmallGroup(128,1302);
// by ID

G=gap.SmallGroup(128,1302);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,456,758,723,184,675,80]);
// Polycyclic

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

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