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G = C42.177D4order 128 = 27

159th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.177D4, C23.468C24, C22.1902- 1+4, (C2×Q8)⋊10Q8, C4.60(C22⋊Q8), C428C4.34C2, C2.19(Q83Q8), C4.75(C4.4D4), (C22×C4).102C23, (C2×C42).569C22, C22.319(C22×D4), C22.109(C22×Q8), (C22×Q8).438C22, C2.C42.204C22, C23.83C23.14C2, C23.67C23.42C2, C2.26(C23.38C23), C2.40(C22.50C24), (C4×C4⋊C4).68C2, (C2×C4×Q8).36C2, (C2×C4⋊Q8).35C2, (C2×C4).56(C2×Q8), (C2×C4).834(C2×D4), C2.36(C2×C22⋊Q8), C2.26(C2×C4.4D4), (C2×C4).827(C4○D4), (C2×C4⋊C4).315C22, C22.344(C2×C4○D4), SmallGroup(128,1300)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.177D4
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.177D4
C1C23 — C42.177D4
C1C23 — C42.177D4
C1C23 — C42.177D4

Generators and relations for C42.177D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 388 in 234 conjugacy classes, 116 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×16], C22 [×3], C22 [×4], C2×C4 [×18], C2×C4 [×36], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×18], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×16], C2.C42 [×18], C2×C42, C2×C42 [×4], C2×C4⋊C4, C2×C4⋊C4 [×8], C4×Q8 [×4], C4⋊Q8 [×4], C22×Q8, C22×Q8 [×2], C4×C4⋊C4, C428C4 [×2], C23.67C23 [×6], C23.83C23 [×4], C2×C4×Q8, C2×C4⋊Q8, C42.177D4
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], C4.4D4 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2- 1+4 [×2], C2×C22⋊Q8, C2×C4.4D4, C23.38C23, C22.50C24 [×2], Q83Q8 [×2], C42.177D4

Smallest permutation representation of C42.177D4
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 20 45 13)(2 17 46 14)(3 18 47 15)(4 19 48 16)(5 37 127 36)(6 38 128 33)(7 39 125 34)(8 40 126 35)(9 42 23 50)(10 43 24 51)(11 44 21 52)(12 41 22 49)(25 54 30 57)(26 55 31 58)(27 56 32 59)(28 53 29 60)(61 65 100 72)(62 66 97 69)(63 67 98 70)(64 68 99 71)(73 108 111 86)(74 105 112 87)(75 106 109 88)(76 107 110 85)(77 113 82 102)(78 114 83 103)(79 115 84 104)(80 116 81 101)(89 117 96 124)(90 118 93 121)(91 119 94 122)(92 120 95 123)
(1 63 55 52)(2 97 56 43)(3 61 53 50)(4 99 54 41)(5 101 75 119)(6 115 76 121)(7 103 73 117)(8 113 74 123)(9 18 65 29)(10 14 66 27)(11 20 67 31)(12 16 68 25)(13 70 26 21)(15 72 28 23)(17 69 32 24)(19 71 30 22)(33 79 85 93)(34 83 86 89)(35 77 87 95)(36 81 88 91)(37 80 106 94)(38 84 107 90)(39 78 108 96)(40 82 105 92)(42 47 100 60)(44 45 98 58)(46 62 59 51)(48 64 57 49)(102 112 120 126)(104 110 118 128)(109 122 127 116)(111 124 125 114)
(1 7 47 127)(2 6 48 126)(3 5 45 125)(4 8 46 128)(9 80 21 83)(10 79 22 82)(11 78 23 81)(12 77 24 84)(13 39 18 36)(14 38 19 35)(15 37 20 34)(16 40 17 33)(25 105 32 85)(26 108 29 88)(27 107 30 87)(28 106 31 86)(41 102 51 115)(42 101 52 114)(43 104 49 113)(44 103 50 116)(53 75 58 111)(54 74 59 110)(55 73 60 109)(56 76 57 112)(61 122 98 117)(62 121 99 120)(63 124 100 119)(64 123 97 118)(65 94 70 89)(66 93 71 92)(67 96 72 91)(68 95 69 90)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111112224
type++++++++--
imageC1C2C2C2C2C2C2D4Q8C4○D42- 1+4
kernelC42.177D4C4×C4⋊C4C428C4C23.67C23C23.83C23C2×C4×Q8C2×C4⋊Q8C42C2×Q8C2×C4C22
# reps112641144122

Matrix representation of C42.177D4 in GL6(𝔽5)

130000
140000
004000
000400
000020
000003
,
420000
410000
001000
000100
000010
000001
,
200000
020000
000100
004000
000001
000010
,
400000
410000
000400
004000
000003
000030

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

C42.177D4 in GAP, Magma, Sage, TeX

C_4^2._{177}D_4
% in TeX

G:=Group("C4^2.177D4");
// GroupNames label

G:=SmallGroup(128,1300);
// by ID

G=gap.SmallGroup(128,1300);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,268,675,80]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^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=b^2*c^-1>;
// generators/relations

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