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G = C42.181D4order 128 = 27

163rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.181D4, C23.484C24, C22.2662+ 1+4, C22.1982- 1+4, C4⋊C421Q8, C4.41(C4⋊Q8), C2.42(D43Q8), C2.22(Q83Q8), (C22×C4).846C23, (C2×C42).578C22, C22.325(C22×D4), C22.119(C22×Q8), (C22×Q8).442C22, C2.68(C22.19C24), C23.81C23.19C2, C23.65C23.60C2, C23.78C23.10C2, C2.C42.218C22, C2.17(C22.31C24), (C4×C4⋊C4).71C2, C2.18(C2×C4⋊Q8), (C2×C4×Q8).39C2, (C2×C4).61(C2×Q8), (C2×C4).365(C2×D4), (C2×C4).157(C4○D4), (C2×C4⋊C4).330C22, C22.360(C2×C4○D4), (C2×C42.C2).22C2, SmallGroup(128,1316)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.181D4
C1C2C22C23C22×C4C2×C42C2×C4×Q8 — C42.181D4
C1C23 — C42.181D4
C1C23 — C42.181D4
C1C23 — C42.181D4

Generators and relations for C42.181D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 372 in 234 conjugacy classes, 120 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×22], C2×C4 [×34], Q8 [×8], C23, C42 [×4], C42 [×6], C4⋊C4 [×8], C4⋊C4 [×30], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×6], C2.C42 [×10], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×16], C4×Q8 [×4], C42.C2 [×4], C22×Q8, C4×C4⋊C4, C23.65C23 [×8], C23.78C23 [×2], C23.81C23 [×2], C2×C4×Q8, C2×C42.C2, C42.181D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ 1+4, 2- 1+4, C22.19C24, C2×C4⋊Q8, C22.31C24, D43Q8 [×2], Q83Q8 [×2], C42.181D4

Smallest permutation representation of C42.181D4
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 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 123 127 91)(34 124 128 92)(35 121 125 89)(36 122 126 90)(65 111 100 87)(66 112 97 88)(67 109 98 85)(68 110 99 86)(73 101 105 77)(74 102 106 78)(75 103 107 79)(76 104 108 80)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)
(1 83 51 101)(2 116 52 78)(3 81 49 103)(4 114 50 80)(5 126 62 66)(6 33 63 98)(7 128 64 68)(8 35 61 100)(9 115 23 77)(10 84 24 102)(11 113 21 79)(12 82 22 104)(13 119 27 73)(14 96 28 106)(15 117 25 75)(16 94 26 108)(17 123 31 85)(18 92 32 110)(19 121 29 87)(20 90 30 112)(34 72 99 40)(36 70 97 38)(37 125 69 65)(39 127 71 67)(41 95 55 105)(42 120 56 74)(43 93 53 107)(44 118 54 76)(45 91 59 109)(46 124 60 86)(47 89 57 111)(48 122 58 88)
(1 111 3 109)(2 86 4 88)(5 74 7 76)(6 105 8 107)(9 87 11 85)(10 110 12 112)(13 65 15 67)(14 99 16 97)(17 77 19 79)(18 104 20 102)(21 123 23 121)(22 90 24 92)(25 127 27 125)(26 36 28 34)(29 113 31 115)(30 84 32 82)(33 55 35 53)(37 75 39 73)(38 106 40 108)(41 100 43 98)(42 68 44 66)(45 101 47 103)(46 80 48 78)(49 91 51 89)(50 122 52 124)(54 126 56 128)(57 81 59 83)(58 116 60 114)(61 93 63 95)(62 120 64 118)(69 117 71 119)(70 96 72 94)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111122244
type++++++++-+-
imageC1C2C2C2C2C2C2D4Q8C4○D42+ 1+42- 1+4
kernelC42.181D4C4×C4⋊C4C23.65C23C23.78C23C23.81C23C2×C4×Q8C2×C42.C2C42C4⋊C4C2×C4C22C22
# reps118221148811

Matrix representation of C42.181D4 in GL6(𝔽5)

030000
200000
004000
000100
000033
000002
,
010000
400000
003000
000200
000040
000004
,
100000
040000
000100
004000
000022
000003
,
400000
010000
000100
001000
000011
000034

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

C42.181D4 in GAP, Magma, Sage, TeX

C_4^2._{181}D_4
% in TeX

G:=Group("C4^2.181D4");
// GroupNames label

G:=SmallGroup(128,1316);
// by ID

G=gap.SmallGroup(128,1316);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,352,675,80]);
// 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*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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