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G = C24.74D4order 128 = 27

29th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.74D4, C4⋊C431D4, C4.62(C4×D4), C4⋊D411C4, C2.5(D4⋊D4), C23.767(C2×D4), (C22×C4).289D4, C4.128(C4⋊D4), C22.4Q1613C2, C22.92C22≀C2, C22.54(C4○D8), (C22×C8).34C22, C22.75(C8⋊C22), C23.80(C22⋊C4), (C23×C4).257C22, (C22×D4).16C22, (C22×C4).1362C23, C2.3(C23.19D4), C2.25(C23.24D4), C2.20(C23.37D4), C2.14(C23.23D4), C22.87(C22.D4), C4⋊C4.70(C2×C4), (C2×D4⋊C4)⋊4C2, (C2×C22⋊C8)⋊15C2, (C2×D4).71(C2×C4), (C2×C4).994(C2×D4), (C2×C4⋊D4).7C2, (C2×C42⋊C2)⋊1C2, (C2×C4).758(C4○D4), (C2×C4⋊C4).764C22, (C2×C4).380(C22×C4), (C22×C4).279(C2×C4), (C2×C4).192(C22⋊C4), C22.266(C2×C22⋊C4), SmallGroup(128,607)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.74D4
C1C2C4C2×C4C22×C4C23×C4C2×C42⋊C2 — C24.74D4
C1C2C2×C4 — C24.74D4
C1C23C23×C4 — C24.74D4
C1C2C2C22×C4 — C24.74D4

Generators and relations for C24.74D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=ac=ca, ad=da, faf=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=be3 >

Subgroups: 476 in 206 conjugacy classes, 64 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×20], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], D4 [×14], C23, C23 [×2], C23 [×14], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C24, C24, C22⋊C8 [×2], D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C23×C4, C22×D4, C22×D4, C22.4Q16 [×2], C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C42⋊C2, C2×C4⋊D4, C24.74D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4○D8 [×2], C8⋊C22 [×2], C23.23D4, C23.24D4, C23.37D4, D4⋊D4 [×2], C23.19D4 [×2], C24.74D4

Smallest permutation representation of C24.74D4
On 64 points
Generators in S64
(1 63)(2 45)(3 57)(4 47)(5 59)(6 41)(7 61)(8 43)(9 22)(10 38)(11 24)(12 40)(13 18)(14 34)(15 20)(16 36)(17 27)(19 29)(21 31)(23 25)(26 39)(28 33)(30 35)(32 37)(42 55)(44 49)(46 51)(48 53)(50 64)(52 58)(54 60)(56 62)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(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)
(2 9)(3 7)(4 15)(6 13)(8 11)(12 16)(17 40)(18 64)(19 38)(20 62)(21 36)(22 60)(23 34)(24 58)(26 56)(27 31)(28 54)(30 52)(32 50)(33 45)(35 43)(37 41)(39 47)(42 61)(44 59)(46 57)(48 63)(51 55)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P4Q4R8A···8H
order12···222224···44···4448···8
size11···144882···24···4884···4

38 irreducible representations

dim1111111222224
type++++++++++
imageC1C2C2C2C2C2C4D4D4D4C4○D4C4○D8C8⋊C22
kernelC24.74D4C22.4Q16C2×C22⋊C8C2×D4⋊C4C2×C42⋊C2C2×C4⋊D4C4⋊D4C4⋊C4C22×C4C24C2×C4C22C22
# reps1212118431482

Matrix representation of C24.74D4 in GL5(𝔽17)

10000
01200
001600
00048
0001313
,
160000
01000
00100
000160
000016
,
10000
016000
001600
00010
00001
,
10000
01000
00100
000160
000016
,
40000
0161500
01100
00006
0001411
,
160000
01000
0161600
00010
0001616

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,2,16,0,0,0,0,0,4,13,0,0,0,8,13],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[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],[4,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,0,14,0,0,0,6,11],[16,0,0,0,0,0,1,16,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16] >;

C24.74D4 in GAP, Magma, Sage, TeX

C_2^4._{74}D_4
% in TeX

G:=Group("C2^4.74D4");
// GroupNames label

G:=SmallGroup(128,607);
// by ID

G=gap.SmallGroup(128,607);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,352,1018,248,2028,124]);
// Polycyclic

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

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