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G = C42.220D4order 128 = 27

202nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.220D4, C42.334C23, (C2×Q8)⋊15Q8, Q8.3(C2×Q8), Q8⋊Q81C2, Q8.Q812C2, C4.Q1618C2, C4⋊C8.42C22, C4⋊C4.41C23, (C2×C8).25C23, C4.29(C22×Q8), C4.Q8.8C22, (C2×C4).276C24, (C22×C4).800D4, C23.658(C2×D4), C4⋊Q8.261C22, C2.D8.79C22, C4.97(C8.C22), C4⋊M4(2).2C2, C4.107(C22⋊Q8), (C2×Q8).365C23, (C4×Q8).297C22, (C22×C4).995C23, (C2×C42).822C22, Q8⋊C4.24C22, C23.38D4.2C2, C22.536(C22×D4), C22.46(C22⋊Q8), C2.19(D8⋊C22), M4(2)⋊C4.10C2, (C2×M4(2)).65C22, C42.C2.103C22, (C22×Q8).474C22, C42⋊C2.117C22, C23.37C23.27C2, (C2×C4×Q8).52C2, C4.86(C2×C4○D4), (C2×C4).100(C2×Q8), C2.57(C2×C22⋊Q8), (C2×C4).1437(C2×D4), C2.23(C2×C8.C22), (C2×C4).293(C4○D4), (C2×C4⋊C4).923C22, SmallGroup(128,1810)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.220D4
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C42.220D4
C1C2C2×C4 — C42.220D4
C1C22C2×C42 — C42.220D4
C1C2C2C2×C4 — C42.220D4

Generators and relations for C42.220D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2c3 >

Subgroups: 300 in 188 conjugacy classes, 102 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×13], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×17], Q8 [×4], Q8 [×10], C23, C42 [×4], C42 [×6], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×5], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C23.38D4 [×2], C4⋊M4(2), M4(2)⋊C4 [×2], Q8⋊Q8 [×2], C4.Q16 [×2], Q8.Q8 [×4], C2×C4×Q8, C23.37C23, C42.220D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C8.C22 [×2], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C8.C22, D8⋊C22, C42.220D4

Smallest permutation representation of C42.220D4
On 64 points
Generators in S64
(1 30 35 47)(2 48 36 31)(3 32 37 41)(4 42 38 25)(5 26 39 43)(6 44 40 27)(7 28 33 45)(8 46 34 29)(9 59 51 23)(10 24 52 60)(11 61 53 17)(12 18 54 62)(13 63 55 19)(14 20 56 64)(15 57 49 21)(16 22 50 58)
(1 37 5 33)(2 34 6 38)(3 39 7 35)(4 36 8 40)(9 53 13 49)(10 50 14 54)(11 55 15 51)(12 52 16 56)(17 63 21 59)(18 60 22 64)(19 57 23 61)(20 62 24 58)(25 48 29 44)(26 45 30 41)(27 42 31 46)(28 47 32 43)
(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 16 35 50)(2 53 36 11)(3 14 37 56)(4 51 38 9)(5 12 39 54)(6 49 40 15)(7 10 33 52)(8 55 34 13)(17 48 61 31)(18 26 62 43)(19 46 63 29)(20 32 64 41)(21 44 57 27)(22 30 58 47)(23 42 59 25)(24 28 60 45)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I···4R4S4T4U4V8A8B8C8D
order1222224···44···444448888
size1111222···24···488888888

32 irreducible representations

dim111111111222244
type+++++++++++--
imageC1C2C2C2C2C2C2C2C2D4D4Q8C4○D4C8.C22D8⋊C22
kernelC42.220D4C23.38D4C4⋊M4(2)M4(2)⋊C4Q8⋊Q8C4.Q16Q8.Q8C2×C4×Q8C23.37C23C42C22×C4C2×Q8C2×C4C4C2
# reps121222411224422

Matrix representation of C42.220D4 in GL6(𝔽17)

0160000
100000
0016000
0001600
0000160
0000016
,
1600000
0160000
004000
0001300
0001340
0080013
,
0130000
1300000
00211015
00141620
001014114
009141115
,
040000
400000
0031150
00211015
00571416
001410156

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,8,0,0,0,13,13,0,0,0,0,0,4,0,0,0,0,0,0,13],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,2,14,10,9,0,0,11,16,14,14,0,0,0,2,1,11,0,0,15,0,14,15],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,3,2,5,14,0,0,1,11,7,10,0,0,15,0,14,15,0,0,0,15,16,6] >;

C42.220D4 in GAP, Magma, Sage, TeX

C_4^2._{220}D_4
% in TeX

G:=Group("C4^2.220D4");
// GroupNames label

G:=SmallGroup(128,1810);
// by ID

G=gap.SmallGroup(128,1810);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,352,2019,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=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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