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G = (C2×D4).304D4order 128 = 27

57th non-split extension by C2×D4 of D4 acting via D4/C22=C2

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

Aliases: (C2×D4).304D4, (C2×Q8).239D4, C4⋊C4.397C23, (C2×C8).313C23, (C2×C4).297C24, (C2×D4).84C23, C23.248(C2×D4), (C2×Q8).72C23, C2.D8.85C22, C4.Q8.14C22, C2.25(D4○SD16), C4⋊D4.25C22, C23.46D44C2, C23.47D44C2, C22⋊C8.19C22, M4(2)⋊C429C2, C22⋊Q8.25C22, C23.36D412C2, C23.20D416C2, C23.19D416C2, C23.24D425C2, (C22×C8).349C22, C22.557(C22×D4), D4⋊C4.181C22, (C22×C4).1013C23, Q8⋊C4.172C22, C4.62(C22.D4), (C2×M4(2)).79C22, C23.33C2311C2, C42⋊C2.126C22, C22.31C24.9C2, C22.24(C22.D4), (C2×C4.Q8)⋊35C2, C4.107(C2×C4○D4), (C2×C4).492(C2×D4), (C22×C8)⋊C211C2, (C2×C4).299(C4○D4), (C2×C4⋊C4).613C22, (C2×C4○D4).142C22, C2.62(C2×C22.D4), SmallGroup(128,1831)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×D4).304D4
C1C2C4C2×C4C22×C4C42⋊C2C23.33C23 — (C2×D4).304D4
C1C2C2×C4 — (C2×D4).304D4
C1C22C2×C4○D4 — (C2×D4).304D4
C1C2C2C2×C4 — (C2×D4).304D4

Generators and relations for (C2×D4).304D4
 G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ebe-1=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ab-1, dcd-1=ece-1=ab2c, ede-1=ab2d3 >

Subgroups: 380 in 199 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×11], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×17], D4 [×12], Q8 [×4], C23, C23 [×2], C23, C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C4○D4 [×8], C22⋊C8 [×4], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×6], C2.D8 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C22⋊Q8, C22×C8, C2×M4(2), C2×C4○D4 [×2], (C22×C8)⋊C2, C23.24D4, C23.36D4, C2×C4.Q8, M4(2)⋊C4, C23.46D4 [×2], C23.19D4 [×2], C23.47D4 [×2], C23.20D4 [×2], C23.33C23, C22.31C24, (C2×D4).304D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4, D4○SD16 [×2], (C2×D4).304D4

Smallest permutation representation of (C2×D4).304D4
On 64 points
Generators in S64
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 56 5 52)(2 25 6 29)(3 50 7 54)(4 27 8 31)(9 60 13 64)(10 35 14 39)(11 62 15 58)(12 37 16 33)(17 59 21 63)(18 34 22 38)(19 61 23 57)(20 36 24 40)(26 41 30 45)(28 43 32 47)(42 55 46 51)(44 49 48 53)
(1 39)(2 58)(3 33)(4 60)(5 35)(6 62)(7 37)(8 64)(9 27)(10 52)(11 29)(12 54)(13 31)(14 56)(15 25)(16 50)(17 26)(18 51)(19 28)(20 53)(21 30)(22 55)(23 32)(24 49)(34 46)(36 48)(38 42)(40 44)(41 63)(43 57)(45 59)(47 61)
(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 12 5 16)(2 24 6 20)(3 10 7 14)(4 22 8 18)(9 46 13 42)(11 44 15 48)(17 43 21 47)(19 41 23 45)(25 62 29 58)(26 39 30 35)(27 60 31 64)(28 37 32 33)(34 55 38 51)(36 53 40 49)(50 57 54 61)(52 63 56 59)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4N4O4P4Q8A8B8C8D8E8F
order12222222244444···4444888888
size11112244822224···4888444488

32 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16
kernel(C2×D4).304D4(C22×C8)⋊C2C23.24D4C23.36D4C2×C4.Q8M4(2)⋊C4C23.46D4C23.19D4C23.47D4C23.20D4C23.33C23C22.31C24C2×D4C2×Q8C2×C4C2
# reps1111112222113184

Matrix representation of (C2×D4).304D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
12160000
750000
0010015
00160161
000101
0010016
,
510000
10120000
004080
00001313
00130130
004440
,
14130000
230000
006600
0014000
001114143
00331414
,
1600000
1010000
004000
00131300
00130130
000004

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,7,0,0,0,0,16,5,0,0,0,0,0,0,1,16,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,15,1,1,16],[5,10,0,0,0,0,1,12,0,0,0,0,0,0,4,0,13,4,0,0,0,0,0,4,0,0,8,13,13,4,0,0,0,13,0,0],[14,2,0,0,0,0,13,3,0,0,0,0,0,0,6,14,11,3,0,0,6,0,14,3,0,0,0,0,14,14,0,0,0,0,3,14],[16,10,0,0,0,0,0,1,0,0,0,0,0,0,4,13,13,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4] >;

(C2×D4).304D4 in GAP, Magma, Sage, TeX

(C_2\times D_4)._{304}D_4
% in TeX

G:=Group("(C2xD4).304D4");
// GroupNames label

G:=SmallGroup(128,1831);
// by ID

G=gap.SmallGroup(128,1831);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,1018,248,4037,1027,124]);
// Polycyclic

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

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