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G = C4211Q8order 128 = 27

11st semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C4211Q8, C23.566C24, C22.2552- (1+4), C22.3402+ (1+4), C428C4.41C2, C425C4.13C2, (C2×C42).630C22, (C22×C4).171C23, C22.140(C22×Q8), (C22×Q8).171C22, C2.55(C22.32C24), C23.81C23.30C2, C23.78C23.17C2, C23.83C23.28C2, C23.67C23.51C2, C23.63C23.39C2, C2.C42.280C22, C23.65C23.70C2, C2.66(C22.36C24), C2.55(C22.33C24), C2.27(C23.41C23), C2.43(C23.37C23), C2.37(C22.35C24), (C4×C4⋊C4).79C2, (C2×C4).136(C2×Q8), (C2×C4).186(C4○D4), (C2×C4⋊C4).387C22, C22.433(C2×C4○D4), SmallGroup(128,1398)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4211Q8
C1C2C22C23C22×C4C2.C42C23.67C23 — C4211Q8
C1C23 — C4211Q8
C1C23 — C4211Q8
C1C23 — C4211Q8

Subgroups: 324 in 184 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C4 [×20], C22 [×7], C2×C4 [×10], C2×C4 [×40], Q8 [×4], C23, C42 [×4], C42 [×2], C4⋊C4 [×18], C22×C4 [×15], C2×Q8 [×4], C2.C42 [×18], C2×C42 [×3], C2×C4⋊C4 [×13], C22×Q8, C4×C4⋊C4, C428C4, C425C4, C23.63C23 [×2], C23.65C23, C23.67C23, C23.78C23 [×2], C23.81C23 [×4], C23.83C23 [×2], C4211Q8

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C23.37C23, C22.32C24, C22.33C24, C22.35C24, C22.36C24, C23.41C23 [×2], C4211Q8

Generators and relations
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c-1 >

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

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

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

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

Matrix representation G ⊆ GL8(𝔽5)

41000000
01000000
00100000
00010000
00002202
00000200
00000430
00000103
,
20000000
02000000
00400000
00040000
00003300
00000200
00000424
00000103
,
40000000
31000000
00300000
00320000
00000010
00000202
00001000
00000103
,
10000000
01000000
00130000
00140000
00003214
00000033
00001331
00004424

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,2,4,1,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,3],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,4,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,3],[4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,3,0,1,4,0,0,0,0,2,0,3,4,0,0,0,0,1,3,3,2,0,0,0,0,4,3,1,4] >;

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4P4Q···4X
order12···244444···44···4
size11···122224···48···8

32 irreducible representations

dim11111111112244
type++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2Q8C4○D42+ (1+4)2- (1+4)
kernelC4211Q8C4×C4⋊C4C428C4C425C4C23.63C23C23.65C23C23.67C23C23.78C23C23.81C23C23.83C23C42C2×C4C22C22
# reps11112112424822

In GAP, Magma, Sage, TeX

C_4^2\rtimes_{11}Q_8
% in TeX

G:=Group("C4^2:11Q8");
// GroupNames label

G:=SmallGroup(128,1398);
// by ID

G=gap.SmallGroup(128,1398);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,352,185,136]);
// Polycyclic

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

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