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G = C42.179D4order 128 = 27

161st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.179D4, C23.481C24, C22.1962- 1+4, (C2×Q8)⋊11Q8, C428C4.37C2, C2.20(Q83Q8), C4.104(C22⋊Q8), (C2×C42).575C22, (C22×C4).111C23, C22.322(C22×D4), C22.116(C22×Q8), (C22×Q8).440C22, C2.66(C22.19C24), C23.78C23.9C2, C23.63C23.29C2, C23.65C23.58C2, C23.83C23.17C2, C23.67C23.43C2, C2.C42.215C22, C2.45(C22.50C24), C2.27(C23.38C23), (C2×C4×Q8).37C2, (C2×C4).58(C2×Q8), C2.39(C2×C22⋊Q8), (C2×C4).1197(C2×D4), (C2×C4).156(C4○D4), (C2×C4⋊C4).327C22, C22.357(C2×C4○D4), SmallGroup(128,1313)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.179D4
C1C2C22C23C22×C4C2×C42C2×C4×Q8 — C42.179D4
C1C23 — C42.179D4
C1C23 — C42.179D4
C1C23 — C42.179D4

Generators and relations for C42.179D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2b2c-1 >

Subgroups: 372 in 234 conjugacy classes, 112 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×18], C2×C4 [×36], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×24], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×10], C2.C42 [×16], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C4×Q8 [×8], C22×Q8 [×2], C428C4, C23.63C23 [×4], C23.65C23 [×2], C23.67C23 [×2], C23.78C23 [×2], C23.83C23 [×2], C2×C4×Q8 [×2], C42.179D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2- 1+4 [×2], C2×C22⋊Q8, C22.19C24, C23.38C23, C22.50C24 [×2], Q83Q8 [×2], C42.179D4

Smallest permutation representation of C42.179D4
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 21 11)(2 16 22 12)(3 13 23 9)(4 14 24 10)(5 56 38 26)(6 53 39 27)(7 54 40 28)(8 55 37 25)(17 30 50 60)(18 31 51 57)(19 32 52 58)(20 29 49 59)(33 74 125 108)(34 75 126 105)(35 76 127 106)(36 73 128 107)(41 72 48 62)(42 69 45 63)(43 70 46 64)(44 71 47 61)(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 74 6 89)(2 107 7 122)(3 76 8 91)(4 105 5 124)(9 127 25 112)(10 34 26 85)(11 125 27 110)(12 36 28 87)(13 35 55 86)(14 126 56 111)(15 33 53 88)(16 128 54 109)(17 118 62 103)(18 95 63 80)(19 120 64 101)(20 93 61 78)(21 108 39 123)(22 73 40 92)(23 106 37 121)(24 75 38 90)(29 98 44 113)(30 67 41 82)(31 100 42 115)(32 65 43 84)(45 81 57 66)(46 114 58 99)(47 83 59 68)(48 116 60 97)(49 119 71 104)(50 96 72 77)(51 117 69 102)(52 94 70 79)
(1 115 21 81)(2 82 22 116)(3 113 23 83)(4 84 24 114)(5 65 38 99)(6 100 39 66)(7 67 40 97)(8 98 37 68)(9 78 13 104)(10 101 14 79)(11 80 15 102)(12 103 16 77)(17 85 50 111)(18 112 51 86)(19 87 52 109)(20 110 49 88)(25 93 55 119)(26 120 56 94)(27 95 53 117)(28 118 54 96)(29 89 59 123)(30 124 60 90)(31 91 57 121)(32 122 58 92)(33 61 125 71)(34 72 126 62)(35 63 127 69)(36 70 128 64)(41 105 48 75)(42 76 45 106)(43 107 46 73)(44 74 47 108)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111112224
type+++++++++--
imageC1C2C2C2C2C2C2C2D4Q8C4○D42- 1+4
kernelC42.179D4C428C4C23.63C23C23.65C23C23.67C23C23.78C23C23.83C23C2×C4×Q8C42C2×Q8C2×C4C22
# reps1142222244122

Matrix representation of C42.179D4 in GL6(𝔽5)

030000
200000
001000
000100
000043
000011
,
040000
100000
004000
000400
000010
000001
,
010000
100000
002000
003300
000030
000022
,
200000
030000
003100
002200
000010
000001

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

C42.179D4 in GAP, Magma, Sage, TeX

C_4^2._{179}D_4
% in TeX

G:=Group("C4^2.179D4");
// GroupNames label

G:=SmallGroup(128,1313);
// by ID

G=gap.SmallGroup(128,1313);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,352,675,80]);
// 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=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations

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