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G = C42.365D4order 128 = 27

61st non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.365D4, C42.721C23, C42(C88D4), C812(C4○D4), C42(C85D4), C85D431C2, C88D456C2, C42(C83Q8), C83Q830C2, (C2×C4)⋊11SD16, (C4×SD16)⋊33C2, C42(C4.4D8), C4.25(C4○D8), C4.4D846C2, C4⋊C4.106C23, (C2×C8).598C23, (C2×C4).365C24, (C4×C8).431C22, C42(C4.SD16), C4.SD1647C2, C4.104(C2×SD16), (C4×D4).87C22, (C22×C4).618D4, C23.393(C2×D4), C4⋊Q8.289C22, (C4×Q8).84C22, C22.3(C2×SD16), (C2×D4).121C23, (C2×Q8).109C23, C2.20(C22×SD16), C4.Q8.161C22, C41D4.154C22, C4⋊D4.170C22, (C22×C8).569C22, C22.625(C22×D4), C22⋊Q8.175C22, D4⋊C4.145C22, (C22×C4).1570C23, (C2×C42).1134C22, Q8⋊C4.137C22, (C2×SD16).149C22, C23.37C2312C2, C22.26C24.38C2, C2.62(C22.26C24), (C2×C4×C8)⋊36C2, C2.34(C2×C4○D8), C4.50(C2×C4○D4), (C2×C4).698(C2×D4), SmallGroup(128,1899)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.365D4
C1C2C4C2×C4C42C4×C8C2×C4×C8 — C42.365D4
C1C2C2×C4 — C42.365D4
C1C2×C4C2×C42 — C42.365D4
C1C2C2C2×C4 — C42.365D4

Generators and relations for C42.365D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 388 in 206 conjugacy classes, 100 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×14], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C4×C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C42, C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C42.C2, C41D4, C4⋊Q8 [×3], C22×C8 [×2], C2×SD16 [×4], C2×C4○D4, C2×C4×C8, C4×SD16 [×4], C88D4 [×4], C4.4D8, C4.SD16, C85D4, C83Q8, C22.26C24, C23.37C23, C42.365D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×SD16 [×6], C4○D8 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C22×SD16, C2×C4○D8, C42.365D4

Smallest permutation representation of C42.365D4
On 64 points
Generators in S64
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 39 21 35)(18 40 22 36)(19 33 23 37)(20 34 24 38)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)
(1 33 27 14)(2 34 28 15)(3 35 29 16)(4 36 30 9)(5 37 31 10)(6 38 32 11)(7 39 25 12)(8 40 26 13)(17 63 42 49)(18 64 43 50)(19 57 44 51)(20 58 45 52)(21 59 46 53)(22 60 47 54)(23 61 48 55)(24 62 41 56)
(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 4 5 8)(2 7 6 3)(9 14 13 10)(11 12 15 16)(17 45 21 41)(18 48 22 44)(19 43 23 47)(20 46 24 42)(25 32 29 28)(26 27 30 31)(33 40 37 36)(34 35 38 39)(49 62 53 58)(50 57 54 61)(51 60 55 64)(52 63 56 59)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8P
order1222222244444···44···48···8
size1111228811112···28···82···2

44 irreducible representations

dim111111111122222
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4SD16C4○D8
kernelC42.365D4C2×C4×C8C4×SD16C88D4C4.4D8C4.SD16C85D4C83Q8C22.26C24C23.37C23C42C22×C4C8C2×C4C4
# reps114411111122888

Matrix representation of C42.365D4 in GL4(𝔽17) generated by

01300
4000
0001
00160
,
4000
0400
0001
00160
,
0100
16000
00512
0055
,
01600
16000
00512
001212
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,16,0,0,1,0],[0,16,0,0,1,0,0,0,0,0,5,5,0,0,12,5],[0,16,0,0,16,0,0,0,0,0,5,12,0,0,12,12] >;

C42.365D4 in GAP, Magma, Sage, TeX

C_4^2._{365}D_4
% in TeX

G:=Group("C4^2.365D4");
// GroupNames label

G:=SmallGroup(128,1899);
// by ID

G=gap.SmallGroup(128,1899);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,184,80,4037,124]);
// Polycyclic

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

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