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G = C42.66D4order 128 = 27

48th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.66D4, C42.142C23, C4.46C4≀C2, (C4×D4).2C4, (C4×Q8).2C4, C22.8C4≀C2, C4⋊D4.9C4, C22⋊Q8.9C4, C42.83(C2×C4), (C4×M4(2))⋊16C2, (C22×C4).665D4, C8⋊C4.142C22, C4(C42.2C22), C4(C42.C22), (C2×C42).186C22, C42.C2.92C22, C23.103(C22⋊C4), C42.C2216C2, C42.2C2215C2, C4.4D4.111C22, C23.36C23.9C2, C2.10(M4(2).8C22), C2.29(C2×C4≀C2), C4⋊C4.23(C2×C4), (C2×C8⋊C4)⋊14C2, (C2×D4).18(C2×C4), (C2×Q8).18(C2×C4), (C2×C4).1170(C2×D4), (C2×C4).95(C22⋊C4), (C2×C4).136(C22×C4), (C22×C4).208(C2×C4), C22.200(C2×C22⋊C4), SmallGroup(128,256)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.66D4
C1C2C22C2×C4C42C2×C42C23.36C23 — C42.66D4
C1C22C2×C4 — C42.66D4
C1C2×C4C2×C42 — C42.66D4
C1C22C22C42 — C42.66D4

Generators and relations for C42.66D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=a2bc3 >

Subgroups: 220 in 117 conjugacy classes, 46 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×5], C8 [×8], C2×C4 [×6], C2×C4 [×10], D4 [×3], Q8, C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], M4(2) [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4 [×4], C8⋊C4, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22×C8, C2×M4(2), C42.C22 [×2], C42.2C22 [×2], C2×C8⋊C4, C4×M4(2), C23.36C23, C42.66D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×4], C2×C22⋊C4, M4(2).8C22, C2×C4≀C2 [×2], C42.66D4

Smallest permutation representation of C42.66D4
On 64 points
Generators in S64
(1 35 5 39)(2 40 6 36)(3 37 7 33)(4 34 8 38)(9 62 13 58)(10 59 14 63)(11 64 15 60)(12 61 16 57)(17 45 21 41)(18 42 22 46)(19 47 23 43)(20 44 24 48)(25 49 29 53)(26 54 30 50)(27 51 31 55)(28 56 32 52)
(1 22 27 57)(2 19 28 62)(3 24 29 59)(4 21 30 64)(5 18 31 61)(6 23 32 58)(7 20 25 63)(8 17 26 60)(9 36 43 52)(10 33 44 49)(11 38 45 54)(12 35 46 51)(13 40 47 56)(14 37 48 53)(15 34 41 50)(16 39 42 55)
(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 17 22 26 27 60 57 8)(2 29 19 59 28 3 62 24)(4 5 21 18 30 31 64 61)(6 25 23 63 32 7 58 20)(9 10 36 33 43 44 52 49)(11 46 38 51 45 12 54 35)(13 14 40 37 47 48 56 53)(15 42 34 55 41 16 50 39)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4J4K4L4M4N4O8A···8P
order122222244444···4444448···8
size111122811112···2448884···4

38 irreducible representations

dim111111111122224
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4≀C2C4≀C2M4(2).8C22
kernelC42.66D4C42.C22C42.2C22C2×C8⋊C4C4×M4(2)C23.36C23C4×D4C4×Q8C4⋊D4C22⋊Q8C42C22×C4C4C22C2
# reps122111222222882

Matrix representation of C42.66D4 in GL4(𝔽17) generated by

01300
4000
0004
0040
,
13000
01300
0001
0010
,
10700
101000
00610
00711
,
10700
7700
00610
00106
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,0,4,0,0,4,0],[13,0,0,0,0,13,0,0,0,0,0,1,0,0,1,0],[10,10,0,0,7,10,0,0,0,0,6,7,0,0,10,11],[10,7,0,0,7,7,0,0,0,0,6,10,0,0,10,6] >;

C42.66D4 in GAP, Magma, Sage, TeX

C_4^2._{66}D_4
% in TeX

G:=Group("C4^2.66D4");
// GroupNames label

G:=SmallGroup(128,256);
// by ID

G=gap.SmallGroup(128,256);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,184,1123,1018,248,1971,102]);
// Polycyclic

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

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