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G = C42.180D4order 128 = 27

162nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.180D4, C23.482C24, C22.2642+ 1+4, C22.1972- 1+4, C2.7Q82, (C2×Q8)⋊12Q8, C429C4.34C2, C2.21(Q83Q8), C4.105(C22⋊Q8), (C22×C4).112C23, (C2×C42).576C22, C22.323(C22×D4), C22.117(C22×Q8), (C22×Q8).441C22, C2.C42.216C22, C23.65C23.59C2, C23.81C23.18C2, C23.67C23.44C2, C2.23(C22.53C24), C2.16(C22.31C24), (C2×C4×Q8).38C2, (C2×C4).59(C2×Q8), (C2×C4).363(C2×D4), C2.40(C2×C22⋊Q8), (C2×C4).897(C4○D4), (C2×C4⋊C4).328C22, C22.358(C2×C4○D4), SmallGroup(128,1314)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.180D4
C1C2C22C23C22×C4C2×C42C2×C4×Q8 — C42.180D4
C1C23 — C42.180D4
C1C23 — C42.180D4
C1C23 — C42.180D4

Generators and relations for C42.180D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 388 in 242 conjugacy classes, 124 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×18], C22 [×3], C22 [×4], C2×C4 [×22], C2×C4 [×34], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×26], C22×C4, C22×C4 [×14], C2×Q8 [×8], C2×Q8 [×8], C2.C42 [×12], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×16], C4×Q8 [×8], C22×Q8 [×2], C429C4, C23.65C23 [×4], C23.67C23 [×4], C23.81C23 [×4], C2×C4×Q8 [×2], C42.180D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C22⋊Q8 [×8], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ 1+4, 2- 1+4, C2×C22⋊Q8 [×2], C22.31C24, Q83Q8 [×2], Q82, C22.53C24, C42.180D4

Smallest permutation representation of C42.180D4
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)
(1 125 28 111)(2 128 25 110)(3 127 26 109)(4 126 27 112)(5 31 116 52)(6 30 113 51)(7 29 114 50)(8 32 115 49)(9 95 71 85)(10 94 72 88)(11 93 69 87)(12 96 70 86)(13 37 58 78)(14 40 59 77)(15 39 60 80)(16 38 57 79)(17 120 64 102)(18 119 61 101)(19 118 62 104)(20 117 63 103)(21 124 98 106)(22 123 99 105)(23 122 100 108)(24 121 97 107)(33 54 76 41)(34 53 73 44)(35 56 74 43)(36 55 75 42)(45 89 67 83)(46 92 68 82)(47 91 65 81)(48 90 66 84)
(1 86 44 103)(2 104 41 87)(3 88 42 101)(4 102 43 85)(5 67 37 98)(6 99 38 68)(7 65 39 100)(8 97 40 66)(9 110 17 76)(10 73 18 111)(11 112 19 74)(12 75 20 109)(13 108 52 81)(14 82 49 105)(15 106 50 83)(16 84 51 107)(21 116 45 78)(22 79 46 113)(23 114 47 80)(24 77 48 115)(25 118 54 93)(26 94 55 119)(27 120 56 95)(28 96 53 117)(29 89 60 124)(30 121 57 90)(31 91 58 122)(32 123 59 92)(33 71 128 64)(34 61 125 72)(35 69 126 62)(36 63 127 70)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111122244
type+++++++-+-
imageC1C2C2C2C2C2D4Q8C4○D42+ 1+42- 1+4
kernelC42.180D4C429C4C23.65C23C23.67C23C23.81C23C2×C4×Q8C42C2×Q8C2×C4C22C22
# reps11444248811

Matrix representation of C42.180D4 in GL6(𝔽5)

010000
400000
001300
001400
000040
000004
,
400000
040000
001300
001400
000040
000004
,
040000
400000
003400
000200
000001
000040
,
030000
200000
002100
000300
000001
000010

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

C42.180D4 in GAP, Magma, Sage, TeX

C_4^2._{180}D_4
% in TeX

G:=Group("C4^2.180D4");
// GroupNames label

G:=SmallGroup(128,1314);
// by ID

G=gap.SmallGroup(128,1314);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,352,675,80]);
// 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,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations

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