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G = C42.224D4order 128 = 27

206th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.224D4, C42.340C23, (C4×Q16)⋊4C2, (C2×C4)⋊11Q16, C4.46(C2×Q16), C42(C4.Q16), C4.Q1649C2, C42(C42Q16), C42Q1648C2, C4⋊C4.57C23, (C4×C8).70C22, Q8.2(C4○D4), C2.9(C22×Q16), C4⋊C8.284C22, (C2×C4).302C24, (C2×C8).147C23, C42(C22⋊Q16), C23.669(C2×D4), (C22×C4).804D4, C4⋊Q8.264C22, C22⋊Q16.5C2, (C4×Q8).70C22, C22.16(C2×Q16), (C2×Q8).373C23, C2.D8.170C22, C42(C23.48D4), C22⋊C8.174C22, (C2×C42).829C22, (C2×Q16).120C22, C23.48D4.5C2, C22.562(C22×D4), C22⋊Q8.165C22, C2.29(D8⋊C22), C42.12C4.34C2, (C22×C4).1018C23, Q8⋊C4.152C22, (C22×Q8).476C22, C2.103(C22.19C24), C23.37C23.28C2, (C2×C4×Q8).53C2, C4.187(C2×C4○D4), (C2×C4).1582(C2×D4), (C2×C4⋊C4).933C22, SmallGroup(128,1836)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.224D4
C1C2C4C2×C4C42C4×Q8C2×C4×Q8 — C42.224D4
C1C2C2×C4 — C42.224D4
C1C2×C4C2×C42 — C42.224D4
C1C2C2C2×C4 — C42.224D4

Generators and relations for C42.224D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=a2c3 >

Subgroups: 308 in 192 conjugacy classes, 98 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×13], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×17], Q8 [×4], Q8 [×12], C23, C42 [×4], C42 [×6], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×4], Q16 [×8], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×6], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8 [×6], C4×Q8 [×3], C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C2×Q16 [×4], C22×Q8, C42.12C4, C4×Q16 [×4], C22⋊Q16 [×2], C42Q16 [×2], C4.Q16 [×2], C23.48D4 [×2], C2×C4×Q8, C23.37C23, C42.224D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×Q16 [×6], C22×D4, C2×C4○D4 [×2], C22.19C24, C22×Q16, D8⋊C22, C42.224D4

Smallest permutation representation of C42.224D4
On 64 points
Generators in S64
(1 7 5 3)(2 58 6 62)(4 60 8 64)(9 15 13 11)(10 37 14 33)(12 39 16 35)(17 23 21 19)(18 49 22 53)(20 51 24 55)(25 44 29 48)(26 32 30 28)(27 46 31 42)(34 40 38 36)(41 47 45 43)(50 56 54 52)(57 63 61 59)
(1 52 63 23)(2 53 64 24)(3 54 57 17)(4 55 58 18)(5 56 59 19)(6 49 60 20)(7 50 61 21)(8 51 62 22)(9 47 34 30)(10 48 35 31)(11 41 36 32)(12 42 37 25)(13 43 38 26)(14 44 39 27)(15 45 40 28)(16 46 33 29)
(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 29 5 25)(2 28 6 32)(3 27 7 31)(4 26 8 30)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 35 21 39)(18 34 22 38)(19 33 23 37)(20 40 24 36)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K···4T4U4V4W4X8A···8H
order12222244444···44···444448···8
size11112211112···24···488884···4

38 irreducible representations

dim11111111122224
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4Q16C4○D4D8⋊C22
kernelC42.224D4C42.12C4C4×Q16C22⋊Q16C42Q16C4.Q16C23.48D4C2×C4×Q8C23.37C23C42C22×C4C2×C4Q8C2
# reps11422221122882

Matrix representation of C42.224D4 in GL4(𝔽17) generated by

16000
2100
0001
00160
,
13000
01300
00016
0010
,
1100
151600
00143
001414
,
161600
0100
00130
0004
G:=sub<GL(4,GF(17))| [16,2,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,0,1,0,0,16,0],[1,15,0,0,1,16,0,0,0,0,14,14,0,0,3,14],[16,0,0,0,16,1,0,0,0,0,13,0,0,0,0,4] >;

C42.224D4 in GAP, Magma, Sage, TeX

C_4^2._{224}D_4
% in TeX

G:=Group("C4^2.224D4");
// GroupNames label

G:=SmallGroup(128,1836);
// by ID

G=gap.SmallGroup(128,1836);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,80,4037,1027,124]);
// Polycyclic

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

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