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G = C42.176D4order 128 = 27

158th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.176D4, C23.467C24, C22.2512+ 1+4, C2.6Q82, C4⋊C420Q8, (C2×Q8)⋊9Q8, C4.40(C4⋊Q8), C4.59(C22⋊Q8), C429C4.31C2, C2.37(D43Q8), (C22×C4).843C23, (C2×C42).568C22, C22.318(C22×D4), C22.108(C22×Q8), (C22×Q8).437C22, C2.26(C22.29C24), C23.78C23.8C2, C23.67C23.41C2, C2.C42.203C22, C23.65C23.55C2, (C4×C4⋊C4).67C2, C2.16(C2×C4⋊Q8), (C2×C4×Q8).35C2, (C2×C4⋊Q8).34C2, (C2×C4).55(C2×Q8), (C2×C4).361(C2×D4), C2.35(C2×C22⋊Q8), (C2×C4).826(C4○D4), (C2×C4⋊C4).876C22, C22.343(C2×C4○D4), SmallGroup(128,1299)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.176D4
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.176D4
C1C23 — C42.176D4
C1C23 — C42.176D4
C1C23 — C42.176D4

Generators and relations for C42.176D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 420 in 250 conjugacy classes, 132 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×26], C2×C4 [×32], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×8], C4⋊C4 [×22], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×12], C2.C42 [×10], C2×C42, C2×C42 [×4], C2×C4⋊C4, C2×C4⋊C4 [×16], C4×Q8 [×4], C4⋊Q8 [×4], C22×Q8, C22×Q8 [×2], C4×C4⋊C4, C429C4 [×2], C23.65C23 [×4], C23.67C23 [×2], C23.78C23 [×4], C2×C4×Q8, C2×C4⋊Q8, C42.176D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×12], C23 [×15], C2×D4 [×6], C2×Q8 [×18], C4○D4 [×2], C24, C22⋊Q8 [×4], C4⋊Q8 [×4], C22×D4, C22×Q8 [×3], C2×C4○D4, 2+ 1+4 [×2], C2×C22⋊Q8, C2×C4⋊Q8, C22.29C24, D43Q8 [×2], Q82 [×2], C42.176D4

Smallest permutation representation of C42.176D4
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 15 44 50)(2 16 41 51)(3 13 42 52)(4 14 43 49)(5 36 37 127)(6 33 38 128)(7 34 39 125)(8 35 40 126)(9 22 17 46)(10 23 18 47)(11 24 19 48)(12 21 20 45)(25 57 54 30)(26 58 55 31)(27 59 56 32)(28 60 53 29)(61 65 72 100)(62 66 69 97)(63 67 70 98)(64 68 71 99)(73 80 111 114)(74 77 112 115)(75 78 109 116)(76 79 110 113)(81 101 108 88)(82 102 105 85)(83 103 106 86)(84 104 107 87)(89 117 124 96)(90 118 121 93)(91 119 122 94)(92 120 123 95)
(1 70 26 10)(2 62 27 17)(3 72 28 12)(4 64 25 19)(5 106 114 122)(6 82 115 90)(7 108 116 124)(8 84 113 92)(9 41 69 56)(11 43 71 54)(13 100 60 21)(14 68 57 48)(15 98 58 23)(16 66 59 46)(18 44 63 55)(20 42 61 53)(22 51 97 32)(24 49 99 30)(29 45 52 65)(31 47 50 67)(33 102 74 118)(34 88 75 96)(35 104 76 120)(36 86 73 94)(37 83 80 91)(38 105 77 121)(39 81 78 89)(40 107 79 123)(85 112 93 128)(87 110 95 126)(101 109 117 125)(103 111 119 127)
(1 125 3 127)(2 128 4 126)(5 50 7 52)(6 49 8 51)(9 102 11 104)(10 101 12 103)(13 37 15 39)(14 40 16 38)(17 85 19 87)(18 88 20 86)(21 83 23 81)(22 82 24 84)(25 110 27 112)(26 109 28 111)(29 114 31 116)(30 113 32 115)(33 43 35 41)(34 42 36 44)(45 106 47 108)(46 105 48 107)(53 73 55 75)(54 76 56 74)(57 79 59 77)(58 78 60 80)(61 94 63 96)(62 93 64 95)(65 122 67 124)(66 121 68 123)(69 118 71 120)(70 117 72 119)(89 100 91 98)(90 99 92 97)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111122224
type+++++++++--+
imageC1C2C2C2C2C2C2C2D4Q8Q8C4○D42+ 1+4
kernelC42.176D4C4×C4⋊C4C429C4C23.65C23C23.67C23C23.78C23C2×C4×Q8C2×C4⋊Q8C42C4⋊C4C2×Q8C2×C4C22
# reps1124241148442

Matrix representation of C42.176D4 in GL6(𝔽5)

300000
020000
000100
004000
000020
000003
,
200000
030000
000100
004000
000010
000001
,
300000
020000
004000
000400
000001
000010
,
010000
400000
000300
003000
000003
000030

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

C42.176D4 in GAP, Magma, Sage, TeX

C_4^2._{176}D_4
% in TeX

G:=Group("C4^2.176D4");
// GroupNames label

G:=SmallGroup(128,1299);
// by ID

G=gap.SmallGroup(128,1299);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,184,675,80]);
// 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,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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