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G = C42.241D4order 128 = 27

223rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.241D4, C42.708C23, C4⋊C4.92C23, (C4×C8).351C22, (C2×C8).458C23, (C2×C4).337C24, C4.SD1641C2, (C22×C4).461D4, C23.679(C2×D4), C4⋊Q8.275C22, (C2×Q8).92C23, C4.78(C4.4D4), C4.20(C8.C22), (C4×M4(2)).31C2, C8⋊C4.169C22, (C2×C42).848C22, C22.597(C22×D4), (C22×C4).1035C23, Q8⋊C4.125C22, C23.38D4.11C2, C22.32(C4.4D4), C42.C2.112C22, (C22×Q8).302C22, C42⋊C2.142C22, C42.30C2218C2, (C2×M4(2)).374C22, C23.37C23.30C2, C4.46(C2×C4○D4), (C2×C4⋊Q8).47C2, (C2×C4).515(C2×D4), C2.48(C2×C4.4D4), C2.39(C2×C8.C22), (C2×C4).301(C4○D4), SmallGroup(128,1871)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.241D4
C1C2C4C2×C4C22×C4C2×M4(2)C4×M4(2) — C42.241D4
C1C2C2×C4 — C42.241D4
C1C22C2×C42 — C42.241D4
C1C2C2C2×C4 — C42.241D4

Generators and relations for C42.241D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2c3 >

Subgroups: 324 in 190 conjugacy classes, 96 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], Q8 [×16], C23, C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×12], C4×C8 [×2], C8⋊C4 [×2], Q8⋊C4 [×16], C2×C42, C2×C4⋊C4 [×2], C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×6], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8 [×2], C4×M4(2), C23.38D4 [×4], C4.SD16 [×4], C42.30C22 [×4], C2×C4⋊Q8, C23.37C23, C42.241D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8.C22 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C8.C22 [×2], C42.241D4

Smallest permutation representation of C42.241D4
On 64 points
Generators in S64
(1 34 25 11)(2 35 26 12)(3 36 27 13)(4 37 28 14)(5 38 29 15)(6 39 30 16)(7 40 31 9)(8 33 32 10)(17 62 46 51)(18 63 47 52)(19 64 48 53)(20 57 41 54)(21 58 42 55)(22 59 43 56)(23 60 44 49)(24 61 45 50)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)
(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 55 25 58)(2 61 26 50)(3 53 27 64)(4 59 28 56)(5 51 29 62)(6 57 30 54)(7 49 31 60)(8 63 32 52)(9 23 40 44)(10 47 33 18)(11 21 34 42)(12 45 35 24)(13 19 36 48)(14 43 37 22)(15 17 38 46)(16 41 39 20)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K···4R8A···8H
order1222224···4444···48···8
size1111222···2448···84···4

32 irreducible representations

dim11111112224
type+++++++++-
imageC1C2C2C2C2C2C2D4D4C4○D4C8.C22
kernelC42.241D4C4×M4(2)C23.38D4C4.SD16C42.30C22C2×C4⋊Q8C23.37C23C42C22×C4C2×C4C4
# reps11444112284

Matrix representation of C42.241D4 in GL6(𝔽17)

400000
13130000
0001600
001000
0000016
000010
,
1600000
0160000
0001600
001000
000001
0000160
,
100000
16160000
0000016
000010
0016000
0001600
,
120000
16160000
000040
0000013
004000
0001300

G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,16,0,0,0],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,4,0,0,0,0,0,0,13,0,0] >;

C42.241D4 in GAP, Magma, Sage, TeX

C_4^2._{241}D_4
% in TeX

G:=Group("C4^2.241D4");
// GroupNames label

G:=SmallGroup(128,1871);
// by ID

G=gap.SmallGroup(128,1871);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,1018,521,2804,172,4037,124]);
// Polycyclic

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

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