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G = C42.231D4order 128 = 27

213rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.231D4, C42.347C23, Q8⋊Q85C2, Q16⋊C46C2, C42Q1622C2, C4⋊C8.50C22, C4⋊C4.66C23, (C2×C8).40C23, Q8.6(C4○D4), C8⋊C4.7C22, (C2×C4).311C24, C23.675(C2×D4), (C22×C4).451D4, C4⋊Q8.268C22, C22⋊Q16.3C2, (C2×Q8).77C23, (C4×Q8).74C22, C4.Q8.16C22, C4.99(C8.C22), C22⋊C8.24C22, C42.6C4.1C2, (C2×Q16).57C22, (C2×C42).838C22, Q8⋊C4.32C22, C23.47D4.1C2, C22.571(C22×D4), C22⋊Q8.171C22, (C22×C4).1027C23, C22.37(C8.C22), (C22×Q8).479C22, C2.112(C22.19C24), C23.37C23.29C2, (C2×C4×Q8).54C2, C4.196(C2×C4○D4), (C2×C4).499(C2×D4), C2.34(C2×C8.C22), (C2×C4⋊C4).939C22, SmallGroup(128,1845)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.231D4
C1C2C4C2×C4C42C4×Q8C2×C4×Q8 — C42.231D4
C1C2C2×C4 — C42.231D4
C1C22C2×C42 — C42.231D4
C1C2C2C2×C4 — C42.231D4

Generators and relations for C42.231D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=a2c3 >

Subgroups: 308 in 190 conjugacy classes, 92 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×14], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×20], 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], C8⋊C4 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×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, C42.6C4, Q16⋊C4 [×4], C22⋊Q16 [×2], C42Q16 [×2], Q8⋊Q8 [×2], C23.47D4 [×2], C2×C4×Q8, C23.37C23, C42.231D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8.C22 [×4], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8.C22 [×2], C42.231D4

Smallest permutation representation of C42.231D4
On 64 points
Generators in S64
(1 7 5 3)(2 58 6 62)(4 60 8 64)(9 15 13 11)(10 33 14 37)(12 35 16 39)(17 55 21 51)(18 24 22 20)(19 49 23 53)(25 42 29 46)(26 32 30 28)(27 44 31 48)(34 40 38 36)(41 47 45 43)(50 56 54 52)(57 63 61 59)
(1 52 63 24)(2 49 64 21)(3 54 57 18)(4 51 58 23)(5 56 59 20)(6 53 60 17)(7 50 61 22)(8 55 62 19)(9 43 38 28)(10 48 39 25)(11 45 40 30)(12 42 33 27)(13 47 34 32)(14 44 35 29)(15 41 36 26)(16 46 37 31)
(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 31 5 27)(2 30 6 26)(3 29 7 25)(4 28 8 32)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 40 21 36)(18 39 22 35)(19 38 23 34)(20 37 24 33)(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)(10,33,14,37)(12,35,16,39)(17,55,21,51)(18,24,22,20)(19,49,23,53)(25,42,29,46)(26,32,30,28)(27,44,31,48)(34,40,38,36)(41,47,45,43)(50,56,54,52)(57,63,61,59), (1,52,63,24)(2,49,64,21)(3,54,57,18)(4,51,58,23)(5,56,59,20)(6,53,60,17)(7,50,61,22)(8,55,62,19)(9,43,38,28)(10,48,39,25)(11,45,40,30)(12,42,33,27)(13,47,34,32)(14,44,35,29)(15,41,36,26)(16,46,37,31), (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), (1,31,5,27)(2,30,6,26)(3,29,7,25)(4,28,8,32)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(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)(10,33,14,37)(12,35,16,39)(17,55,21,51)(18,24,22,20)(19,49,23,53)(25,42,29,46)(26,32,30,28)(27,44,31,48)(34,40,38,36)(41,47,45,43)(50,56,54,52)(57,63,61,59), (1,52,63,24)(2,49,64,21)(3,54,57,18)(4,51,58,23)(5,56,59,20)(6,53,60,17)(7,50,61,22)(8,55,62,19)(9,43,38,28)(10,48,39,25)(11,45,40,30)(12,42,33,27)(13,47,34,32)(14,44,35,29)(15,41,36,26)(16,46,37,31), (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), (1,31,5,27)(2,30,6,26)(3,29,7,25)(4,28,8,32)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(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),(10,33,14,37),(12,35,16,39),(17,55,21,51),(18,24,22,20),(19,49,23,53),(25,42,29,46),(26,32,30,28),(27,44,31,48),(34,40,38,36),(41,47,45,43),(50,56,54,52),(57,63,61,59)], [(1,52,63,24),(2,49,64,21),(3,54,57,18),(4,51,58,23),(5,56,59,20),(6,53,60,17),(7,50,61,22),(8,55,62,19),(9,43,38,28),(10,48,39,25),(11,45,40,30),(12,42,33,27),(13,47,34,32),(14,44,35,29),(15,41,36,26),(16,46,37,31)], [(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)], [(1,31,5,27),(2,30,6,26),(3,29,7,25),(4,28,8,32),(9,51,13,55),(10,50,14,54),(11,49,15,53),(12,56,16,52),(17,40,21,36),(18,39,22,35),(19,38,23,34),(20,37,24,33),(41,64,45,60),(42,63,46,59),(43,62,47,58),(44,61,48,57)])

32 conjugacy classes

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

32 irreducible representations

dim11111111122244
type+++++++++++--
imageC1C2C2C2C2C2C2C2C2D4D4C4○D4C8.C22C8.C22
kernelC42.231D4C42.6C4Q16⋊C4C22⋊Q16C42Q16Q8⋊Q8C23.47D4C2×C4×Q8C23.37C23C42C22×C4Q8C4C22
# reps11422221122822

Matrix representation of C42.231D4 in GL6(𝔽17)

100000
13160000
000100
0016000
000001
0000160
,
1300000
0130000
001002
0001150
0001160
00160016
,
15160000
320000
00131693
00113149
005341
00145164
,
15160000
320000
00151458
00142812
00861514
0069142

G:=sub<GL(6,GF(17))| [1,13,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,16,0,0,0,1,1,0,0,0,0,15,16,0,0,0,2,0,0,16],[15,3,0,0,0,0,16,2,0,0,0,0,0,0,13,1,5,14,0,0,16,13,3,5,0,0,9,14,4,16,0,0,3,9,1,4],[15,3,0,0,0,0,16,2,0,0,0,0,0,0,15,14,8,6,0,0,14,2,6,9,0,0,5,8,15,14,0,0,8,12,14,2] >;

C42.231D4 in GAP, Magma, Sage, TeX

C_4^2._{231}D_4
% in TeX

G:=Group("C4^2.231D4");
// GroupNames label

G:=SmallGroup(128,1845);
// by ID

G=gap.SmallGroup(128,1845);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,521,80,4037,1027,124]);
// 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,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations

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