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G = C42.109D4order 128 = 27

91st non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.109D4, C23.10M4(2), (C2×C8)⋊24D4, C24.63(C2×C4), C4.49(C41D4), C2.21(C89D4), (C22×D4).32C4, C22.180(C4×D4), C4.204(C4⋊D4), C4.89(C4.4D4), C22.56(C8○D4), (C22×C8).55C22, (C23×C4).25C22, (C2×C42).314C22, C23.317(C22×C4), C22.70(C2×M4(2)), C2.19(C24.4C4), (C22×C4).1638C23, C2.7(C24.3C22), (C2×C4⋊C8)⋊16C2, (C2×C4×D4).23C2, (C2×C4⋊C4).61C4, (C2×C8⋊C4)⋊26C2, (C2×C22⋊C8)⋊38C2, (C2×C4).1544(C2×D4), (C2×C22⋊C4).41C4, (C2×C4).944(C4○D4), (C22×C4).128(C2×C4), (C2×C4).135(C22⋊C4), C22.277(C2×C22⋊C4), C2.28((C22×C8)⋊C2), SmallGroup(128,687)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.109D4
C1C2C4C2×C4C22×C4C2×C42C2×C8⋊C4 — C42.109D4
C1C23 — C42.109D4
C1C22×C4 — C42.109D4
C1C2C2C22×C4 — C42.109D4

Generators and relations for C42.109D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 364 in 192 conjugacy classes, 68 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×20], C8 [×6], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×10], C22×C4 [×3], C22×C4 [×2], C22×C4 [×12], C2×D4 [×6], C24 [×2], C8⋊C4 [×2], C22⋊C8 [×8], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C22×C8 [×4], C23×C4 [×2], C22×D4, C2×C8⋊C4, C2×C22⋊C8 [×4], C2×C4⋊C8, C2×C4×D4, C42.109D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×M4(2) [×2], C8○D4 [×2], C24.3C22, C24.4C4, (C22×C8)⋊C2, C89D4 [×4], C42.109D4

Smallest permutation representation of C42.109D4
On 64 points
Generators in S64
(1 43 33 24)(2 48 34 21)(3 45 35 18)(4 42 36 23)(5 47 37 20)(6 44 38 17)(7 41 39 22)(8 46 40 19)(9 61 49 31)(10 58 50 28)(11 63 51 25)(12 60 52 30)(13 57 53 27)(14 62 54 32)(15 59 55 29)(16 64 56 26)
(1 27 5 31)(2 28 6 32)(3 29 7 25)(4 30 8 26)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 64)
(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 5 55)(2 17 6 21)(3 53 7 49)(4 19 8 23)(9 35 13 39)(10 58 14 62)(11 37 15 33)(12 60 16 64)(18 31 22 27)(20 25 24 29)(26 52 30 56)(28 54 32 50)(34 44 38 48)(36 46 40 42)(41 57 45 61)(43 59 47 63)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···144441···14···44···4

44 irreducible representations

dim1111111122222
type+++++++
imageC1C2C2C2C2C4C4C4D4D4C4○D4M4(2)C8○D4
kernelC42.109D4C2×C8⋊C4C2×C22⋊C8C2×C4⋊C8C2×C4×D4C2×C22⋊C4C2×C4⋊C4C22×D4C42C2×C8C2×C4C23C22
# reps1141142244488

Matrix representation of C42.109D4 in GL6(𝔽17)

100000
010000
000100
0016000
000040
0000213
,
100000
010000
001000
000100
0000130
0000013
,
1130000
9160000
000100
0016000
0000915
000008
,
1600000
810000
000100
001000
0000130
0000013

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,9,0,0,0,0,13,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,15,8],[16,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13] >;

C42.109D4 in GAP, Magma, Sage, TeX

C_4^2._{109}D_4
% in TeX

G:=Group("C4^2.109D4");
// GroupNames label

G:=SmallGroup(128,687);
// by ID

G=gap.SmallGroup(128,687);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,124]);
// Polycyclic

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

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