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G = C42.66D4  order 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.C22⋊16C2, C42.2C22⋊15C2, 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

Derived series
C1 — C2×C4 — C42.66D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C23.36C23 — C42.66D4
Lower central
C1 — C22 — C2×C4 — C42.66D4
Upper central
C1 — C2×C4 — C2×C42 — C42.66D4
Jennings
C1 — C22 — C22 — C42 — 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, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C8⋊C4, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×M4(2), C42.C22, C42.2C22, C2×C8⋊C4, C4×M4(2), C23.36C23, C42.66D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, M4(2).8C22, C2×C4≀C2, C42.66D4

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

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

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

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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