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G = C232Q16order 128 = 27

1st semidirect product of C23 and Q16 acting via Q16/C4=C22

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

Aliases: C232Q16, C24.81D4, C4⋊C4.78D4, (C2×C8).40D4, (C2×Q8).77D4, C4.62C22≀C2, (C22×Q16)⋊1C2, C4.15(C4⋊D4), C4.28(C41D4), C23.898(C2×D4), (C22×C4).139D4, C22.52(C2×Q16), C22.4Q1637C2, C2.12(C8.D4), C2.6(C232D4), C22.194C22≀C2, C2.28(D4.7D4), C2.12(C8.18D4), C22.102(C4○D8), (C22×C8).106C22, (C23×C4).269C22, C2.18(C22⋊Q16), (C22×Q8).47C22, C22.219(C4⋊D4), (C22×C4).1432C23, C22.117(C8.C22), (C2×Q8⋊C4)⋊12C2, (C2×C4).1022(C2×D4), (C2×C22⋊C8).26C2, (C2×C22⋊Q8).12C2, (C2×C4).615(C4○D4), (C2×C4⋊C4).103C22, SmallGroup(128,733)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C232Q16
C1C2C22C23C22×C4C23×C4C2×C22⋊Q8 — C232Q16
C1C2C22×C4 — C232Q16
C1C23C23×C4 — C232Q16
C1C2C2C22×C4 — C232Q16

Generators and relations for C232Q16
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=d4, ab=ba, eae-1=ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 400 in 196 conjugacy classes, 56 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C8 [×3], C2×C4 [×2], C2×C4 [×4], C2×C4 [×23], Q8 [×12], C23, C23 [×2], C23 [×6], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×5], Q16 [×8], C22×C4 [×2], C22×C4 [×10], C2×Q8 [×4], C2×Q8 [×10], C24, C22⋊C8 [×2], Q8⋊C4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22⋊Q8 [×8], C22×C8 [×2], C2×Q16 [×6], C23×C4, C22×Q8 [×2], C22.4Q16, C2×C22⋊C8, C2×Q8⋊C4 [×2], C2×C22⋊Q8 [×2], C22×Q16, C232Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, Q16 [×2], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×Q16, C4○D8, C8.C22 [×2], C232D4, C22⋊Q16 [×2], D4.7D4 [×2], C8.18D4, C8.D4, C232Q16

Smallest permutation representation of C232Q16
On 64 points
Generators in S64
(1 5)(2 57)(3 7)(4 59)(6 61)(8 63)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)(25 54)(26 39)(27 56)(28 33)(29 50)(30 35)(31 52)(32 37)(34 42)(36 44)(38 46)(40 48)(41 49)(43 51)(45 53)(47 55)(58 62)(60 64)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)
(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 33 5 37)(2 40 6 36)(3 39 7 35)(4 38 8 34)(9 44 13 48)(10 43 14 47)(11 42 15 46)(12 41 16 45)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(49 64 53 60)(50 63 54 59)(51 62 55 58)(52 61 56 57)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N8A···8H
order12···2224444444···48···8
size11···1442222448···84···4

32 irreducible representations

dim111111222222224
type+++++++++++--
imageC1C2C2C2C2C2D4D4D4D4D4C4○D4Q16C4○D8C8.C22
kernelC232Q16C22.4Q16C2×C22⋊C8C2×Q8⋊C4C2×C22⋊Q8C22×Q16C4⋊C4C2×C8C22×C4C2×Q8C24C2×C4C23C22C22
# reps111221421412442

Matrix representation of C232Q16 in GL6(𝔽17)

1600000
0160000
001000
0081600
0000160
0000141
,
100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
001000
000100
0000160
0000016
,
1430000
14140000
009200
0011800
000080
00001515
,
7160000
16100000
001000
000100
0000315
0000514

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

C232Q16 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2Q_{16}
% in TeX

G:=Group("C2^3:2Q16");
// GroupNames label

G:=SmallGroup(128,733);
// by ID

G=gap.SmallGroup(128,733);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,2019,1018,248]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,e*a*e^-1=a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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