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G = C42.67D4order 128 = 27

49th non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.67D4, C42.148C23, C4.15C4≀C2, C4⋊Q8.16C4, C41D4.11C4, C42.89(C2×C4), (C22×C4).738D4, C42.6C435C2, C8⋊C4.146C22, C42.C229C2, (C2×C42).192C22, C22.2(C4.D4), C23.104(C22⋊C4), C4.4D4.115C22, C2.30(C42⋊C22), C22.26C24.10C2, C2.35(C2×C4≀C2), (C2×C4○D4).4C4, (C2×C8⋊C4)⋊15C2, (C2×D4).22(C2×C4), (C2×Q8).22(C2×C4), (C2×C4).1176(C2×D4), C2.12(C2×C4.D4), (C22×C4).214(C2×C4), (C2×C4).142(C22×C4), (C2×C4).320(C22⋊C4), C22.206(C2×C22⋊C4), SmallGroup(128,262)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.67D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.67D4
C1C22C2×C4 — C42.67D4
C1C22C2×C42 — C42.67D4
C1C22C22C42 — C42.67D4

Generators and relations for C42.67D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 284 in 128 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×8], C8 [×6], C2×C4 [×6], C2×C4 [×11], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×4], C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8, C2×C4○D4 [×2], C42.C22 [×4], C2×C8⋊C4, C42.6C4, C22.26C24, C42.67D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C2×C4.D4, C2×C4≀C2, C42⋊C22, C42.67D4

Smallest permutation representation of C42.67D4
On 64 points
Generators in S64
(1 45 19 34)(2 39 20 42)(3 47 21 36)(4 33 22 44)(5 41 23 38)(6 35 24 46)(7 43 17 40)(8 37 18 48)(9 56 63 30)(10 27 64 53)(11 50 57 32)(12 29 58 55)(13 52 59 26)(14 31 60 49)(15 54 61 28)(16 25 62 51)
(1 58 23 16)(2 63 24 13)(3 60 17 10)(4 57 18 15)(5 62 19 12)(6 59 20 9)(7 64 21 14)(8 61 22 11)(25 45 55 38)(26 42 56 35)(27 47 49 40)(28 44 50 37)(29 41 51 34)(30 46 52 39)(31 43 53 36)(32 48 54 33)
(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)
(1 37 58 28 23 44 16 50)(2 31 63 43 24 53 13 36)(3 46 60 52 17 39 10 30)(4 55 57 38 18 25 15 45)(5 33 62 32 19 48 12 54)(6 27 59 47 20 49 9 40)(7 42 64 56 21 35 14 26)(8 51 61 34 22 29 11 41)

G:=sub<Sym(64)| (1,45,19,34)(2,39,20,42)(3,47,21,36)(4,33,22,44)(5,41,23,38)(6,35,24,46)(7,43,17,40)(8,37,18,48)(9,56,63,30)(10,27,64,53)(11,50,57,32)(12,29,58,55)(13,52,59,26)(14,31,60,49)(15,54,61,28)(16,25,62,51), (1,58,23,16)(2,63,24,13)(3,60,17,10)(4,57,18,15)(5,62,19,12)(6,59,20,9)(7,64,21,14)(8,61,22,11)(25,45,55,38)(26,42,56,35)(27,47,49,40)(28,44,50,37)(29,41,51,34)(30,46,52,39)(31,43,53,36)(32,48,54,33), (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), (1,37,58,28,23,44,16,50)(2,31,63,43,24,53,13,36)(3,46,60,52,17,39,10,30)(4,55,57,38,18,25,15,45)(5,33,62,32,19,48,12,54)(6,27,59,47,20,49,9,40)(7,42,64,56,21,35,14,26)(8,51,61,34,22,29,11,41)>;

G:=Group( (1,45,19,34)(2,39,20,42)(3,47,21,36)(4,33,22,44)(5,41,23,38)(6,35,24,46)(7,43,17,40)(8,37,18,48)(9,56,63,30)(10,27,64,53)(11,50,57,32)(12,29,58,55)(13,52,59,26)(14,31,60,49)(15,54,61,28)(16,25,62,51), (1,58,23,16)(2,63,24,13)(3,60,17,10)(4,57,18,15)(5,62,19,12)(6,59,20,9)(7,64,21,14)(8,61,22,11)(25,45,55,38)(26,42,56,35)(27,47,49,40)(28,44,50,37)(29,41,51,34)(30,46,52,39)(31,43,53,36)(32,48,54,33), (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), (1,37,58,28,23,44,16,50)(2,31,63,43,24,53,13,36)(3,46,60,52,17,39,10,30)(4,55,57,38,18,25,15,45)(5,33,62,32,19,48,12,54)(6,27,59,47,20,49,9,40)(7,42,64,56,21,35,14,26)(8,51,61,34,22,29,11,41) );

G=PermutationGroup([(1,45,19,34),(2,39,20,42),(3,47,21,36),(4,33,22,44),(5,41,23,38),(6,35,24,46),(7,43,17,40),(8,37,18,48),(9,56,63,30),(10,27,64,53),(11,50,57,32),(12,29,58,55),(13,52,59,26),(14,31,60,49),(15,54,61,28),(16,25,62,51)], [(1,58,23,16),(2,63,24,13),(3,60,17,10),(4,57,18,15),(5,62,19,12),(6,59,20,9),(7,64,21,14),(8,61,22,11),(25,45,55,38),(26,42,56,35),(27,47,49,40),(28,44,50,37),(29,41,51,34),(30,46,52,39),(31,43,53,36),(32,48,54,33)], [(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)], [(1,37,58,28,23,44,16,50),(2,31,63,43,24,53,13,36),(3,46,60,52,17,39,10,30),(4,55,57,38,18,25,15,45),(5,33,62,32,19,48,12,54),(6,27,59,47,20,49,9,40),(7,42,64,56,21,35,14,26),(8,51,61,34,22,29,11,41)])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L8A···8H8I8J8K8L
order122222224···444448···88888
size111122882···244884···48888

32 irreducible representations

dim1111111122244
type++++++++
imageC1C2C2C2C2C4C4C4D4D4C4≀C2C4.D4C42⋊C22
kernelC42.67D4C42.C22C2×C8⋊C4C42.6C4C22.26C24C41D4C4⋊Q8C2×C4○D4C42C22×C4C4C22C2
# reps1411122422822

Matrix representation of C42.67D4 in GL6(𝔽17)

400000
040000
000010
00441615
0016000
0000013
,
010000
100000
0013000
0001300
0000130
0000013
,
670000
10110000
0033015
00531515
0055015
001271411
,
1170000
7110000
00141408
0012399
00121209
0050140

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,16,0,0,0,0,4,0,0,0,0,1,16,0,0,0,0,0,15,0,13],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[6,10,0,0,0,0,7,11,0,0,0,0,0,0,3,5,5,12,0,0,3,3,5,7,0,0,0,15,0,14,0,0,15,15,15,11],[11,7,0,0,0,0,7,11,0,0,0,0,0,0,14,12,12,5,0,0,14,3,12,0,0,0,0,9,0,14,0,0,8,9,9,0] >;

C42.67D4 in GAP, Magma, Sage, TeX

C_4^2._{67}D_4
% in TeX

G:=Group("C4^2.67D4");
// GroupNames label

G:=SmallGroup(128,262);
// by ID

G=gap.SmallGroup(128,262);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,352,1123,1018,248,1971,102]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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