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G = C42.212D4order 128 = 27

194th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.212D4, C42.322C23, (C2×Q8)⋊28D4, Q8.4(C2×D4), C4⋊SD161C2, C42Q1618C2, C4⋊C8.38C22, C44(C8.C22), (C2×C8).12C23, C4.68(C22×D4), C4⋊C4.378C23, C4⋊M4(2)⋊3C2, (C2×C4).241C24, Q8.D412C2, (C2×D4).50C23, C23.653(C2×D4), (C22×C4).794D4, C4⋊Q8.256C22, C4.169(C4⋊D4), C23.36D46C2, (C4×Q8).291C22, (C2×Q16).51C22, (C2×Q8).358C23, (C2×SD16).3C22, D4⋊C4.20C22, C41D4.137C22, C22.33(C4⋊D4), (C2×C42).810C22, (C22×C4).971C23, Q8⋊C4.22C22, C22.501(C22×D4), C2.12(D8⋊C22), C4.4D4.126C22, (C2×M4(2)).48C22, (C22×Q8).470C22, C22.26C24.30C2, (C2×C4×Q8)⋊36C2, C4.151(C2×C4○D4), C2.59(C2×C4⋊D4), (C2×C4).1420(C2×D4), (C2×C8.C22)⋊14C2, C2.15(C2×C8.C22), (C2×C4).272(C4○D4), (C2×C4⋊C4).921C22, (C2×C4○D4).116C22, SmallGroup(128,1769)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.212D4
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C42.212D4
C1C2C2×C4 — C42.212D4
C1C22C2×C42 — C42.212D4
C1C2C2C2×C4 — C42.212D4

Generators and relations for C42.212D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, cbc-1=b-1, bd=db, dcd=c3 >

Subgroups: 444 in 242 conjugacy classes, 102 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×19], D4 [×12], Q8 [×4], Q8 [×10], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×8], C2×Q8 [×3], C4○D4 [×8], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×SD16 [×4], C2×Q16 [×4], C8.C22 [×8], C22×Q8, C2×C4○D4 [×2], C23.36D4 [×2], C4⋊M4(2), C4⋊SD16 [×2], C42Q16 [×2], Q8.D4 [×4], C2×C4×Q8, C22.26C24, C2×C8.C22 [×2], C42.212D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8.C22 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8.C22, D8⋊C22, C42.212D4

Smallest permutation representation of C42.212D4
On 64 points
Generators in S64
(1 54 29 61)(2 62 30 55)(3 56 31 63)(4 64 32 49)(5 50 25 57)(6 58 26 51)(7 52 27 59)(8 60 28 53)(9 24 40 44)(10 45 33 17)(11 18 34 46)(12 47 35 19)(13 20 36 48)(14 41 37 21)(15 22 38 42)(16 43 39 23)
(1 14 5 10)(2 11 6 15)(3 16 7 12)(4 13 8 9)(17 61 21 57)(18 58 22 62)(19 63 23 59)(20 60 24 64)(25 33 29 37)(26 38 30 34)(27 35 31 39)(28 40 32 36)(41 50 45 54)(42 55 46 51)(43 52 47 56)(44 49 48 53)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 45)(18 48)(19 43)(20 46)(21 41)(22 44)(23 47)(24 42)(26 28)(27 31)(30 32)(34 36)(35 39)(38 40)(49 62)(50 57)(51 60)(52 63)(53 58)(54 61)(55 64)(56 59)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111111222244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4C8.C22D8⋊C22
kernelC42.212D4C23.36D4C4⋊M4(2)C4⋊SD16C42Q16Q8.D4C2×C4×Q8C22.26C24C2×C8.C22C42C22×C4C2×Q8C2×C4C4C2
# reps121224112224422

Matrix representation of C42.212D4 in GL6(𝔽17)

1150000
1160000
000040
0044139
004000
0000013
,
1600000
0160000
00111615
0000160
000100
00101616
,
100000
1160000
001212010
0012121010
005500
00127510
,
100000
1160000
001000
0001600
0000160
0001611

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

C42.212D4 in GAP, Magma, Sage, TeX

C_4^2._{212}D_4
% in TeX

G:=Group("C4^2.212D4");
// GroupNames label

G:=SmallGroup(128,1769);
// by ID

G=gap.SmallGroup(128,1769);
# by ID

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

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

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