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G = C42.230D4order 128 = 27

212nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.230D4, C42.346C23, Q83(C4○D4), Q8⋊D45C2, C4⋊SD165C2, C4.Q1622C2, C4⋊C4.65C23, C4⋊C8.49C22, (C2×C8).39C23, SD16⋊C47C2, C8⋊C4.6C22, (C2×C4).310C24, C42.6C44C2, (C2×D4).91C23, (C4×D4).77C22, C23.674(C2×D4), (C22×C4).450D4, C4⋊Q8.267C22, C2.D8.87C22, C22.D815C2, C4.98(C8.C22), C22⋊C8.23C22, (C4×Q8).302C22, (C2×Q8).377C23, D4⋊C4.31C22, C4⋊D4.166C22, C41D4.141C22, C22.48(C8⋊C22), (C2×C42).837C22, Q8⋊C4.31C22, (C2×SD16).12C22, C22.570(C22×D4), (C22×C4).1026C23, (C22×Q8).478C22, C22.26C24.32C2, C2.111(C22.19C24), (C2×C4×Q8)⋊39C2, C4.195(C2×C4○D4), (C2×C4).498(C2×D4), C2.34(C2×C8⋊C22), C2.33(C2×C8.C22), (C2×C4⋊C4).938C22, SmallGroup(128,1844)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.230D4
C1C2C4C2×C4C42C4×Q8C2×C4×Q8 — C42.230D4
C1C2C2×C4 — C42.230D4
C1C22C2×C42 — C42.230D4
C1C2C2C2×C4 — C42.230D4

Generators and relations for C42.230D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=dbd=a2b, dcd=c3 >

Subgroups: 404 in 212 conjugacy classes, 92 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×22], D4 [×12], Q8 [×4], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4×Q8 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×SD16 [×4], C22×Q8, C2×C4○D4, C42.6C4, SD16⋊C4 [×4], Q8⋊D4 [×2], C4⋊SD16 [×2], C4.Q16 [×2], C22.D8 [×2], C2×C4×Q8, C22.26C24, C42.230D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22, C2×C8.C22, C42.230D4

Smallest permutation representation of C42.230D4
On 64 points
Generators in S64
(1 23 5 19)(2 41 6 45)(3 17 7 21)(4 43 8 47)(9 56 13 52)(10 30 14 26)(11 50 15 54)(12 32 16 28)(18 62 22 58)(20 64 24 60)(25 38 29 34)(27 40 31 36)(33 51 37 55)(35 53 39 49)(42 61 46 57)(44 63 48 59)
(1 28 59 51)(2 25 60 56)(3 30 61 53)(4 27 62 50)(5 32 63 55)(6 29 64 52)(7 26 57 49)(8 31 58 54)(9 45 34 24)(10 42 35 21)(11 47 36 18)(12 44 37 23)(13 41 38 20)(14 46 39 17)(15 43 40 22)(16 48 33 19)
(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)(2 40)(3 35)(4 38)(5 33)(6 36)(7 39)(8 34)(9 58)(10 61)(11 64)(12 59)(13 62)(14 57)(15 60)(16 63)(17 30)(18 25)(19 28)(20 31)(21 26)(22 29)(23 32)(24 27)(41 54)(42 49)(43 52)(44 55)(45 50)(46 53)(47 56)(48 51)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4R4S4T8A8B8C8D
order122222224···44···4448888
size111122882···24···4888888

32 irreducible representations

dim11111111122244
type+++++++++++-+
imageC1C2C2C2C2C2C2C2C2D4D4C4○D4C8.C22C8⋊C22
kernelC42.230D4C42.6C4SD16⋊C4Q8⋊D4C4⋊SD16C4.Q16C22.D8C2×C4×Q8C22.26C24C42C22×C4Q8C4C22
# reps11422221122822

Matrix representation of C42.230D4 in GL6(𝔽17)

16150000
010000
0016200
0016100
0000162
0000161
,
400000
040000
0000115
0000116
0011500
0011600
,
1390000
440000
0000107
000050
0071000
0012000
,
1390000
440000
0000160
0000161
0016000
0016100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,1,0,0,0,0,15,16,0,0],[13,4,0,0,0,0,9,4,0,0,0,0,0,0,0,0,7,12,0,0,0,0,10,0,0,0,10,5,0,0,0,0,7,0,0,0],[13,4,0,0,0,0,9,4,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,16,16,0,0,0,0,0,1,0,0] >;

C42.230D4 in GAP, Magma, Sage, TeX

C_4^2._{230}D_4
% in TeX

G:=Group("C4^2.230D4");
// GroupNames label

G:=SmallGroup(128,1844);
// by ID

G=gap.SmallGroup(128,1844);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,521,80,4037,1027,124]);
// Polycyclic

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

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