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G = C23.37D8order 128 = 27

8th non-split extension by C23 of D8 acting via D8/D4=C2

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

Aliases: C23.37D8, C24.158D4, C23.19Q16, C22⋊C87C4, C4⋊C4.297D4, C4.136(C4×D4), C22.36(C2×D8), C221(C2.D8), C4.1(C22⋊Q8), C2.2(C22⋊D8), (C22×C4).48Q8, C23.71(C4⋊C4), (C22×C4).282D4, C23.757(C2×D4), C22.29(C2×Q16), C22.4Q1633C2, C22.77C22≀C2, C2.2(C22⋊Q16), C22.67(C8⋊C22), (C22×C8).101C22, (C23×C4).248C22, C2.2(C22.D8), C23.7Q8.12C2, C2.8(C23.8Q8), (C22×C4).1349C23, C2.2(C23.48D4), C22.56(C8.C22), C2.10(M4(2)⋊C4), C22.81(C22.D4), (C2×C8)⋊4(C2×C4), (C2×C2.D8)⋊2C2, C2.9(C2×C2.D8), (C2×C4).51(C4⋊C4), (C2×C4).979(C2×D4), (C2×C4).199(C2×Q8), (C2×C4⋊C4).51C22, (C22×C4⋊C4).15C2, (C2×C22⋊C8).25C2, C22.109(C2×C4⋊C4), (C2×C4).745(C4○D4), (C2×C4).548(C22×C4), (C22×C4).271(C2×C4), SmallGroup(128,584)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.37D8
C1C2C22C23C22×C4C23×C4C22×C4⋊C4 — C23.37D8
C1C2C2×C4 — C23.37D8
C1C23C23×C4 — C23.37D8
C1C2C2C22×C4 — C23.37D8

Generators and relations for C23.37D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 356 in 180 conjugacy classes, 72 normal (28 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×8], C22 [×7], C22 [×4], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×32], C23, C23 [×6], C23 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×16], C24, C2.C42, C22⋊C8 [×4], C2.D8 [×4], C2×C22⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×5], C22×C8 [×2], C23×C4, C23×C4, C22.4Q16 [×2], C23.7Q8, C2×C22⋊C8, C2×C2.D8 [×2], C22×C4⋊C4, C23.37D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2.D8 [×4], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×D8, C2×Q16, C8⋊C22, C8.C22, C23.8Q8, C2×C2.D8, M4(2)⋊C4, C22⋊D8, C22⋊Q16, C22.D8, C23.48D4, C23.37D8

Smallest permutation representation of C23.37D8
On 64 points
Generators in S64
(1 51)(2 61)(3 53)(4 63)(5 55)(6 57)(7 49)(8 59)(9 21)(10 43)(11 23)(12 45)(13 17)(14 47)(15 19)(16 41)(18 29)(20 31)(22 25)(24 27)(26 44)(28 46)(30 48)(32 42)(33 60)(34 52)(35 62)(36 54)(37 64)(38 56)(39 58)(40 50)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(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 30 51 19)(2 29 52 18)(3 28 53 17)(4 27 54 24)(5 26 55 23)(6 25 56 22)(7 32 49 21)(8 31 50 20)(9 58 42 39)(10 57 43 38)(11 64 44 37)(12 63 45 36)(13 62 46 35)(14 61 47 34)(15 60 48 33)(16 59 41 40)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H
order12···2222244444···444448···8
size11···1222222224···488884···4

38 irreducible representations

dim1111111222222244
type++++++++-++-+-
imageC1C2C2C2C2C2C4D4D4Q8D4C4○D4D8Q16C8⋊C22C8.C22
kernelC23.37D8C22.4Q16C23.7Q8C2×C22⋊C8C2×C2.D8C22×C4⋊C4C22⋊C8C4⋊C4C22×C4C22×C4C24C2×C4C23C23C22C22
# reps1211218412144411

Matrix representation of C23.37D8 in GL5(𝔽17)

10000
016000
001600
00010
0001616
,
10000
01000
00100
000160
000016
,
160000
016000
001600
00010
00001
,
160000
014300
0141400
00012
000016
,
40000
016700
07100
0001615
00001

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

C23.37D8 in GAP, Magma, Sage, TeX

C_2^3._{37}D_8
% in TeX

G:=Group("C2^3.37D8");
// GroupNames label

G:=SmallGroup(128,584);
// by ID

G=gap.SmallGroup(128,584);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,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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