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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:C8:7C4, C4:C4.297D4, C4.136(C4xD4), C22.36(C2xD8), C22:1(C2.D8), C4.1(C22:Q8), C2.2(C22:D8), (C22xC4).48Q8, C23.71(C4:C4), (C22xC4).282D4, C23.757(C2xD4), C22.29(C2xQ16), C22.4Q16:33C2, C22.77C22wrC2, C2.2(C22:Q16), C22.67(C8:C22), (C22xC8).101C22, (C23xC4).248C22, C2.2(C22.D8), C23.7Q8.12C2, C2.8(C23.8Q8), (C22xC4).1349C23, C2.2(C23.48D4), C22.56(C8.C22), C2.10(M4(2):C4), C22.81(C22.D4), (C2xC8):4(C2xC4), (C2xC2.D8):2C2, C2.9(C2xC2.D8), (C2xC4).51(C4:C4), (C2xC4).979(C2xD4), (C2xC4).199(C2xQ8), (C2xC4:C4).51C22, (C22xC4:C4).15C2, (C2xC22:C8).25C2, C22.109(C2xC4:C4), (C2xC4).745(C4oD4), (C2xC4).548(C22xC4), (C22xC4).271(C2xC4), SmallGroup(128,584)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C23.37D8
C1C2C22C23C22xC4C23xC4C22xC4:C4 — C23.37D8
C1C2C2xC4 — C23.37D8
C1C23C23xC4 — C23.37D8
C1C2C2C22xC4 — 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, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, C23, C23, C23, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C24, C2.C42, C22:C8, C2.D8, C2xC22:C4, C2xC4:C4, C2xC4:C4, C2xC4:C4, C22xC8, C23xC4, C23xC4, C22.4Q16, C23.7Q8, C2xC22:C8, C2xC2.D8, C22xC4:C4, C23.37D8
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C4:C4, D8, Q16, C22xC4, C2xD4, C2xQ8, C4oD4, C2.D8, C2xC4:C4, C4xD4, C22wrC2, C22:Q8, C22.D4, C2xD8, C2xQ16, C8:C22, C8.C22, C23.8Q8, C2xC2.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 19)(2 61)(3 21)(4 63)(5 23)(6 57)(7 17)(8 59)(9 62)(10 22)(11 64)(12 24)(13 58)(14 18)(15 60)(16 20)(25 46)(26 37)(27 48)(28 39)(29 42)(30 33)(31 44)(32 35)(34 51)(36 53)(38 55)(40 49)(41 56)(43 50)(45 52)(47 54)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)
(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 27 19 38)(2 26 20 37)(3 25 21 36)(4 32 22 35)(5 31 23 34)(6 30 24 33)(7 29 17 40)(8 28 18 39)(9 53 62 46)(10 52 63 45)(11 51 64 44)(12 50 57 43)(13 49 58 42)(14 56 59 41)(15 55 60 48)(16 54 61 47)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111222222244
type++++++++-++-+-
imageC1C2C2C2C2C2C4D4D4Q8D4C4oD4D8Q16C8:C22C8.C22
kernelC23.37D8C22.4Q16C23.7Q8C2xC22:C8C2xC2.D8C22xC4:C4C22:C8C4:C4C22xC4C22xC4C24C2xC4C23C23C22C22
# 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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