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G = C42.125D4order 128 = 27

107th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.125D4, (C2×C8)⋊8Q8, C4⋊Q826C4, C2.5(C8⋊Q8), C4.15(C4×Q8), C42.167(C2×C4), C23.816(C2×D4), (C22×C4).304D4, C4.83(C22⋊Q8), C22.38(C4⋊Q8), C428C4.13C2, C22.4Q16.52C2, (C22×C8).411C22, (C2×C42).337C22, C22.103(C8⋊C22), (C22×C4).1427C23, C22.71(C4.4D4), C22.92(C8.C22), C2.25(C23.37D4), C2.25(C23.38D4), C2.5(C42.28C22), C2.15(C23.67C23), (C2×C4⋊Q8).16C2, C4⋊C4.100(C2×C4), (C2×C4).214(C2×Q8), (C2×C8⋊C4).36C2, (C2×C4).1367(C2×D4), (C2×C4⋊C4).98C22, (C2×C4).610(C4○D4), (C2×C4).441(C22×C4), (C2×C4).142(C22⋊C4), C22.302(C2×C22⋊C4), SmallGroup(128,725)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.125D4
C1C2C22C23C22×C4C2×C42C2×C8⋊C4 — C42.125D4
C1C2C2×C4 — C42.125D4
C1C23C2×C42 — C42.125D4
C1C2C2C22×C4 — C42.125D4

Generators and relations for C42.125D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 276 in 138 conjugacy classes, 64 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], Q8 [×8], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C2.C42 [×2], C8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×4], C428C4, C2×C8⋊C4, C2×C4⋊Q8, C42.125D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C8⋊C22 [×2], C8.C22 [×2], C23.67C23, C23.37D4, C23.38D4, C42.28C22 [×2], C8⋊Q8 [×2], C42.125D4

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

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim111111222244
type++++++-++-
imageC1C2C2C2C2C4D4Q8D4C4○D4C8⋊C22C8.C22
kernelC42.125D4C22.4Q16C428C4C2×C8⋊C4C2×C4⋊Q8C4⋊Q8C42C2×C8C22×C4C2×C4C22C22
# reps141118242422

Matrix representation of C42.125D4 in GL8(𝔽17)

115000000
116000000
0016150000
00110000
00000010
000041130
00001000
0000012316
,
160000000
016000000
001600000
000160000
000001600
00001000
0000131640
0000614613
,
712000000
1310000000
009150000
00780000
000016787
00000666
0000151187
000021434
,
815000000
79000000
009150000
00780000
000021434
00009021
0000811913
00000666

G:=sub<GL(8,GF(17))| [1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,12,0,0,0,0,1,13,0,3,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,13,6,0,0,0,0,16,0,16,14,0,0,0,0,0,0,4,6,0,0,0,0,0,0,0,13],[7,13,0,0,0,0,0,0,12,10,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,15,8,0,0,0,0,0,0,0,0,16,0,15,2,0,0,0,0,7,6,11,14,0,0,0,0,8,6,8,3,0,0,0,0,7,6,7,4],[8,7,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,15,8,0,0,0,0,0,0,0,0,2,9,8,0,0,0,0,0,14,0,11,6,0,0,0,0,3,2,9,6,0,0,0,0,4,1,13,6] >;

C42.125D4 in GAP, Magma, Sage, TeX

C_4^2._{125}D_4
% in TeX

G:=Group("C4^2.125D4");
// GroupNames label

G:=SmallGroup(128,725);
// by ID

G=gap.SmallGroup(128,725);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,723,436,2019,248]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,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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