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G = C42.431D4order 128 = 27

64th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.431D4, C4⋊Q816C4, C4.9(C4×D4), (C2×C8).229D4, (C2×C4).37Q16, C42(Q8⋊C4), C2.2(C85D4), (C2×C4).72SD16, C4.72(C4⋊D4), C42.265(C2×C4), C2.2(C4⋊Q16), C23.795(C2×D4), (C22×C4).579D4, C22.39(C2×Q16), C429C4.11C2, C2.3(C4.SD16), C22.70(C2×SD16), C22.31(C41D4), (C22×C8).489C22, (C22×Q8).32C22, (C22×C4).1400C23, (C2×C42).1071C22, C22.62(C4.4D4), C2.8(C24.3C22), (C2×C4×C8).23C2, (C2×C4⋊Q8).9C2, (C2×C4).734(C2×D4), (C2×Q8).88(C2×C4), (C2×C4⋊C4).83C22, (C2×Q8⋊C4).7C2, C2.22(C2×Q8⋊C4), (C2×C4).591(C4○D4), (C2×C4).414(C22×C4), (C2×C4).255(C22⋊C4), C22.278(C2×C22⋊C4), SmallGroup(128,688)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.431D4
C1C2C22C23C22×C4C2×C42C2×C4×C8 — C42.431D4
C1C2C2×C4 — C42.431D4
C1C23C2×C42 — C42.431D4
C1C2C2C22×C4 — C42.431D4

Generators and relations for C42.431D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 308 in 166 conjugacy classes, 80 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×6], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×14], Q8 [×12], C23, C42 [×4], C4⋊C4 [×16], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C4×C8 [×2], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8 [×2], C429C4, C2×C4×C8, C2×Q8⋊C4 [×4], C2×C4⋊Q8, C42.431D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], SD16 [×4], Q16 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], Q8⋊C4 [×8], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×SD16 [×2], C2×Q16 [×2], C24.3C22, C2×Q8⋊C4 [×2], C4.SD16 [×2], C85D4, C4⋊Q16, C42.431D4

Smallest permutation representation of C42.431D4
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 39 29 13)(2 40 30 14)(3 37 31 15)(4 38 32 16)(5 120 108 19)(6 117 105 20)(7 118 106 17)(8 119 107 18)(9 49 35 25)(10 50 36 26)(11 51 33 27)(12 52 34 28)(21 110 122 126)(22 111 123 127)(23 112 124 128)(24 109 121 125)(41 81 71 57)(42 82 72 58)(43 83 69 59)(44 84 70 60)(45 53 67 61)(46 54 68 62)(47 55 65 63)(48 56 66 64)(73 103 113 85)(74 104 114 86)(75 101 115 87)(76 102 116 88)(77 89 93 99)(78 90 94 100)(79 91 95 97)(80 92 96 98)
(1 87 9 79)(2 86 10 78)(3 85 11 77)(4 88 12 80)(5 84 23 66)(6 83 24 65)(7 82 21 68)(8 81 22 67)(13 75 25 91)(14 74 26 90)(15 73 27 89)(16 76 28 92)(17 72 126 62)(18 71 127 61)(19 70 128 64)(20 69 125 63)(29 101 35 95)(30 104 36 94)(31 103 33 93)(32 102 34 96)(37 113 51 99)(38 116 52 98)(39 115 49 97)(40 114 50 100)(41 111 53 119)(42 110 54 118)(43 109 55 117)(44 112 56 120)(45 107 57 123)(46 106 58 122)(47 105 59 121)(48 108 60 124)
(1 127 29 111)(2 126 30 110)(3 125 31 109)(4 128 32 112)(5 52 108 28)(6 51 105 27)(7 50 106 26)(8 49 107 25)(9 18 35 119)(10 17 36 118)(11 20 33 117)(12 19 34 120)(13 22 39 123)(14 21 40 122)(15 24 37 121)(16 23 38 124)(41 115 71 75)(42 114 72 74)(43 113 69 73)(44 116 70 76)(45 79 67 95)(46 78 68 94)(47 77 65 93)(48 80 66 96)(53 97 61 91)(54 100 62 90)(55 99 63 89)(56 98 64 92)(57 87 81 101)(58 86 82 104)(59 85 83 103)(60 88 84 102)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4T8A···8P
order12···24···44···48···8
size11···12···28···82···2

44 irreducible representations

dim111111222222
type++++++++-
imageC1C2C2C2C2C4D4D4D4SD16Q16C4○D4
kernelC42.431D4C429C4C2×C4×C8C2×Q8⋊C4C2×C4⋊Q8C4⋊Q8C42C2×C8C22×C4C2×C4C2×C4C2×C4
# reps111418242884

Matrix representation of C42.431D4 in GL5(𝔽17)

10000
00100
016000
000160
000016
,
160000
01000
00100
00040
000013
,
40000
013600
06400
000015
00080
,
160000
041100
0111300
00004
00040

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,13],[4,0,0,0,0,0,13,6,0,0,0,6,4,0,0,0,0,0,0,8,0,0,0,15,0],[16,0,0,0,0,0,4,11,0,0,0,11,13,0,0,0,0,0,0,4,0,0,0,4,0] >;

C42.431D4 in GAP, Magma, Sage, TeX

C_4^2._{431}D_4
% in TeX

G:=Group("C4^2.431D4");
// GroupNames label

G:=SmallGroup(128,688);
// by ID

G=gap.SmallGroup(128,688);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248]);
// Polycyclic

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

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