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G = Q324C4order 128 = 27

3rd semidirect product of Q32 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: Q324C4, C42.133D4, C16.8(C2×C4), C4.29(C4×D4), C2.17(C4×D8), (C2×C8).211D4, (C2×C4).110D8, Q16.3(C2×C4), (C2×Q32).6C2, C165C4.1C2, C164C4.2C2, C4.15(C4○D8), C8.42(C4○D4), C8.39(C22×C4), (C4×Q16).19C2, C22.65(C2×D8), (C4×C8).219C22, (C2×C8).506C23, (C2×C16).20C22, C2.Q32.6C2, C2.5(Q32⋊C2), C2.D8.154C22, (C2×Q16).105C22, (C2×C4).772(C2×D4), SmallGroup(128,908)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — Q324C4
C1C2C4C2×C4C2×C8C4×C8C4×Q16 — Q324C4
C1C2C4C8 — Q324C4
C1C22C42C4×C8 — Q324C4
C1C2C2C2C2C4C4C2×C8 — Q324C4

Generators and relations for Q324C4
 G = < a,b,c | a16=c4=1, b2=a8, bab-1=a-1, cac-1=a9, bc=cb >

Subgroups: 148 in 74 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×8], C22, C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×6], C16 [×2], C16, C42, C42 [×2], C4⋊C4 [×4], C2×C8 [×2], Q16 [×4], Q16 [×2], C2×Q8 [×2], C4×C8, Q8⋊C4 [×2], C2.D8 [×2], C2×C16 [×2], Q32 [×4], C4×Q8 [×2], C2×Q16 [×2], C165C4, C2.Q32 [×2], C164C4, C4×Q16 [×2], C2×Q32, Q324C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, Q32⋊C2 [×2], Q324C4

Smallest permutation representation of Q324C4
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)
(1 120 9 128)(2 119 10 127)(3 118 11 126)(4 117 12 125)(5 116 13 124)(6 115 14 123)(7 114 15 122)(8 113 16 121)(17 46 25 38)(18 45 26 37)(19 44 27 36)(20 43 28 35)(21 42 29 34)(22 41 30 33)(23 40 31 48)(24 39 32 47)(49 79 57 71)(50 78 58 70)(51 77 59 69)(52 76 60 68)(53 75 61 67)(54 74 62 66)(55 73 63 65)(56 72 64 80)(81 97 89 105)(82 112 90 104)(83 111 91 103)(84 110 92 102)(85 109 93 101)(86 108 94 100)(87 107 95 99)(88 106 96 98)
(1 102 28 72)(2 111 29 65)(3 104 30 74)(4 97 31 67)(5 106 32 76)(6 99 17 69)(7 108 18 78)(8 101 19 71)(9 110 20 80)(10 103 21 73)(11 112 22 66)(12 105 23 75)(13 98 24 68)(14 107 25 77)(15 100 26 70)(16 109 27 79)(33 62 118 82)(34 55 119 91)(35 64 120 84)(36 57 121 93)(37 50 122 86)(38 59 123 95)(39 52 124 88)(40 61 125 81)(41 54 126 90)(42 63 127 83)(43 56 128 92)(44 49 113 85)(45 58 114 94)(46 51 115 87)(47 60 116 96)(48 53 117 89)

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

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

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

32 conjugacy classes

class 1 2A2B2C4A···4F4G···4N8A8B8C8D8E8F16A···16H
order12224···44···488888816···16
size11112···28···82222444···4

32 irreducible representations

dim1111111222224
type+++++++++-
imageC1C2C2C2C2C2C4D4D4C4○D4D8C4○D8Q32⋊C2
kernelQ324C4C165C4C2.Q32C164C4C4×Q16C2×Q32Q32C42C2×C8C8C2×C4C4C2
# reps1121218112444

Matrix representation of Q324C4 in GL6(𝔽17)

1410000
730000
001015167
002101016
0016772
001016157
,
1600000
1110000
0010100
001700
0000101
000017
,
400000
040000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [14,7,0,0,0,0,1,3,0,0,0,0,0,0,10,2,16,10,0,0,15,10,7,16,0,0,16,10,7,15,0,0,7,16,2,7],[16,11,0,0,0,0,0,1,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,0,0,0,0,10,1,0,0,0,0,1,7],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

Q324C4 in GAP, Magma, Sage, TeX

Q_{32}\rtimes_4C_4
% in TeX

G:=Group("Q32:4C4");
// GroupNames label

G:=SmallGroup(128,908);
// by ID

G=gap.SmallGroup(128,908);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1430,100,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^16=c^4=1,b^2=a^8,b*a*b^-1=a^-1,c*a*c^-1=a^9,b*c=c*b>;
// generators/relations

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