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G = C42.169D4order 128 = 27

151st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.169D4, C23.446C24, C22.1792- 1+4, C4⋊C417Q8, C43(C4⋊Q8), C2.23(D4×Q8), (C2×Q8).168D4, C4.60(C4⋊D4), C2.15(Q83Q8), C22.98(C22×Q8), (C2×C42).551C22, (C22×C4).837C23, C22.297(C22×D4), (C22×Q8).434C22, C2.C42.184C22, C23.67C23.38C2, C23.65C23.51C2, C23.81C23.13C2, C2.23(C23.38C23), (C4×C4⋊C4).62C2, C2.11(C2×C4⋊Q8), (C2×C4×Q8).33C2, (C2×C4).74(C2×D4), (C2×C4⋊Q8).31C2, (C2×C4).49(C2×Q8), C2.38(C2×C4⋊D4), (C2×C4).894(C4○D4), (C2×C4⋊C4).869C22, C22.323(C2×C4○D4), SmallGroup(128,1278)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.169D4
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.169D4
C1C23 — C42.169D4
C1C23 — C42.169D4
C1C23 — C42.169D4

Generators and relations for C42.169D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 436 in 270 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 [×30], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×16], C2.C42 [×10], C2×C42, C2×C42 [×4], C2×C4⋊C4, C2×C4⋊C4 [×16], C4×Q8 [×4], C4⋊Q8 [×12], C22×Q8, C22×Q8 [×2], C4×C4⋊C4, C23.65C23 [×4], C23.67C23 [×2], C23.81C23 [×4], C2×C4×Q8, C2×C4⋊Q8, C2×C4⋊Q8 [×2], C42.169D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×8], C23 [×15], C2×D4 [×12], C2×Q8 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C4⋊Q8 [×4], C22×D4 [×2], C22×Q8 [×2], C2×C4○D4, 2- 1+4 [×2], C2×C4⋊D4, C2×C4⋊Q8, C23.38C23, D4×Q8 [×2], Q83Q8 [×2], C42.169D4

Smallest permutation representation of C42.169D4
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 36 44 127)(2 35 41 126)(3 34 42 125)(4 33 43 128)(5 50 37 15)(6 49 38 14)(7 52 39 13)(8 51 40 16)(9 87 17 104)(10 86 18 103)(11 85 19 102)(12 88 20 101)(21 108 45 81)(22 107 46 84)(23 106 47 83)(24 105 48 82)(25 74 54 112)(26 73 55 111)(27 76 56 110)(28 75 53 109)(29 78 60 116)(30 77 57 115)(31 80 58 114)(32 79 59 113)(61 117 72 96)(62 120 69 95)(63 119 70 94)(64 118 71 93)(65 89 100 124)(66 92 97 123)(67 91 98 122)(68 90 99 121)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111122224
type++++++++-+-
imageC1C2C2C2C2C2C2D4Q8D4C4○D42- 1+4
kernelC42.169D4C4×C4⋊C4C23.65C23C23.67C23C23.81C23C2×C4×Q8C2×C4⋊Q8C42C4⋊C4C2×Q8C2×C4C22
# reps114241348442

Matrix representation of C42.169D4 in GL6(𝔽5)

430000
110000
004000
000400
000001
000040
,
430000
110000
001000
000100
000010
000001
,
100000
010000
000100
004000
000010
000004
,
300000
220000
000100
001000
000010
000004

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

C42.169D4 in GAP, Magma, Sage, TeX

C_4^2._{169}D_4
% in TeX

G:=Group("C4^2.169D4");
// GroupNames label

G:=SmallGroup(128,1278);
// by ID

G=gap.SmallGroup(128,1278);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,184,675,80]);
// 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^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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